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✓ 67 results · all verified · 28 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 39 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Abelian Varieties, Base Change, and Arithmetic Models

1 · Prerequisites

2 · Summary

This page develops the theory of abelian varieties over a field together with their arithmetic models over a discrete valuation ring. It constructs the dual abelian variety and the Poincare bundle, develops theta groups, Mumford maps and polarizations with their square degrees, and then builds the Neron model of an abelian variety over an arbitrary discrete valuation ring, proving the Neron-Ogg-Shafarevich criterion in residue characteristics prime to the torsion prime and good reduction, conditional coherent cohomology base change, and prime-to-residue-characteristic torsion specialization for abelian schemes.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passOpen item page →

Group schemes over a base scheme

Definition

Let $S$ be a scheme. An $S$ -group scheme is a group object in the category of $S$ -schemes: an $S$ -scheme $G$ , with structure morphism $p : G \to S$ , together with $S$ -morphisms $m : G \times_{S} G \to G, e : S \to G, i : G \to G$ called multiplication, unit section and inverse, such that the following identities of $S$ -morphisms hold. The fibre product is the one of Fibre product of schemes and exists by Existence of all scheme fibre products; the canonical identifications $S \times_{S} G ≅ G ≅ G \times_{S} S$ used below are those of Uniqueness of the fibre product.

(Associativity) $m \circ (m \times_{S} {id}_{G}) = m \circ ({id}_{G} \times_{S} m)$ as morphisms $G \times_{S} G \times_{S} G \to G$ .

(Unit laws) $m \circ (e \times_{S} {id}_{G}) = {id}_{G}$ and $m \circ ({id}_{G} \times_{S} e) = {id}_{G}$ under the canonical identifications above.

(Inverse laws) $m \circ ({id}_{G}, i) = e \circ p$ and $m \circ (i, {id}_{G}) = e \circ p$ as morphisms $G \to G$ , where $({id}_{G}, i) : G \to G \times_{S} G$ and $(i, {id}_{G}) : G \to G \times_{S} G$ are induced by the universal property of the fibre product.

A morphism of $S$ -group schemes $f : G \to H$ is an $S$ -morphism with $f \circ m_{G} = m_{H} \circ (f \times_{S} f)$ , $f \circ e_{G} = e_{H}$ and $f \circ i_{G} = i_{H} \circ f$ , where $f \times_{S} f : G \times_{S} G \to H \times_{S} H$ is the induced morphism.

Functor of points. For an $S$ -scheme $T$ put $G (T) = {Hom}_{S} (T, G)$ . The composites $T \overset{Δ}{\to} T \times_{S} T \overset{α \times_{S} β}{\to} G \times_{S} G \overset{m}{\to} G$ , $T \to S \overset{e}{\to} G$ and $i \circ α$ make $G (T)$ a group, and this structure is natural in $T$ by the universal property of the fibre product. A morphism $f : G \to H$ of $S$ -group schemes induces homomorphisms $G (T) \to H (T)$ compatible with the transition maps of $T$ . For $S = Spec k$ and $G$ of finite type this is the notion of Group schemes of finite type over a field, and the group object diagrams above are equivalent to the requirement that each $G (T)$ be a group naturally in $T$ . The definition is a condition on given data and quotes no new existence statement.

Commutativity. $G$ is commutative when $m = m \circ σ$ , where $σ : G \times_{S} G \to G \times_{S} G$ is the exchange isomorphism; equivalently, each $G (T)$ is abelian.

Closed subgroup schemes. A closed subgroup scheme of $G$ is a closed immersion $j : H \to G$ for which there exist $S$ -morphisms $m_{H} : H \times_{S} H \to H$ , $e_{H} : S \to H$ , $i_{H} : H \to H$ with $j \circ m_{H} = m \circ (j \times_{S} j)$ , $j \circ e_{H} = e$ and $j \circ i_{H} = i \circ j$ ; since $j$ is a monomorphism these morphisms are unique if they exist, $H$ becomes an $S$ -group scheme and $j$ a morphism of $S$ -group schemes. A closed subgroup scheme $H$ is normal when $H (T)$ is a normal subgroup of $G (T)$ for every $S$ -scheme $T$ ; equivalently, the conjugation morphism $G \times_{S} H \to G$ , $(α, β) \mapsto m_{G} (m_{G} (α, β), i_{G} (α))$ , factors through $j$ .

Kernels. For a morphism $f : G \to H$ of $S$ -group schemes, the kernel is the fibre product $K = G \times_{H} S$ formed with $f$ and the unit section $e_{H} : S \to H$ , so that $K \to G$ is the base change of $e_{H}$ along $f$ and is a closed immersion whenever $e_{H}$ is one, in particular whenever $H$ is separated over $S$ ; the induced $S$ -morphisms make $K$ an $S$ -group scheme, and for every $S$ -scheme $T$ the sequence $0 \to K (T) \to G (T) \to H (T)$ is exact.

Base change. For any morphism $S^{'} \to S$ , the base change $G_{S^{'}} = G \times_{S} S^{'}$ of Base change of objects, morphisms and properties carries an induced $S^{'}$ -group structure with $m_{S^{'}}, e_{S^{'}}, i_{S^{'}}$ obtained from $m, e, i$ under the canonical isomorphism $G_{S^{'}} \times_{S^{'}} G_{S^{'}} ≅ (G \times_{S} G) \times_{S} S^{'}$ . It satisfies $G_{S^{'}} (T) = G (T)$ for every $S^{'}$ -scheme $T$ , regarded as an $S$ -scheme via $T \to S^{'} \to S$ ; this is the base-change convention used for all group schemes on this page. In particular a morphism $f : G \to H$ of $S$ -group schemes base changes to a morphism $f_{S^{'}}$ of $S^{'}$ -group schemes.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passOpen item page →

Finite etale schemes over a complete local ring and splitting

Statement

Assume AC. Let $R$ be a Noetherian local ring which is complete and separated for its maximal ideal $m$ , with residue field $k$ , and let $Y \to Spec R$ be a finite etale $R$ -scheme. Then the reduction map $Y (R) \to Y (k)$ is a bijection. If in addition $k$ has no nontrivial finite separable field extension, then every finite etale $R$ -algebra of rank $d$ is isomorphic to $R^{d}$ as an $R$ -algebra, and $Y$ is a disjoint union of $d$ copies of $Spec R$ .

Facts & Assumptions

Given: AC, a complete separated Noetherian local ring $(R, m)$ with residue field $k$ , and a finite etale $R$ -scheme $Y$ .

[F1]

Reduction $A \mapsto A / m A$ is an equivalence between finite etale $R$ -algebras and finite etale $k$ -algebras, for $(R, m)$ complete and separated; more generally for a nilpotent ideal $I$ in a commutative ring, reduction gives such an equivalence (Finite étale algebras over a complete local ring are determined by reduction, Finite étale algebras lift uniquely through nilpotent thickenings, both assuming AC).

[F2]

A module-finite commutative $A$ -algebra $D$ is finite etale over $A$ if and only if $D$ is finitely presented and flat as an $A$ -module with $Ω_{D / A} = 0$ ; then $D$ is finite projective locally free, its rank is locally constant and equals the number of geometric points in a fibre (Finite étale algebras have finite locally free underlying modules, assuming AC).

[F3]

At a point of a locally finite-type morphism whose stalk of relative differentials vanishes, the residue-field extension is finite separable; in particular a finite etale field extension $L / k$ , viewed as $Spec L \to Spec k$ , has $L / k$ finite separable (Unramified residue extensions are finite separable, assuming AC).

[F4]

A commutative Artinian ring is the product of its localizations at its finitely many maximal ideals, and a local Artinian ring which is a domain is a field; a regular local ring is a domain (An Artinian ring is canonically the finite product of its localizations at its maximal ideals, An Artinian integral domain is a field, regular local rings are domains and cohen macaulay).

[F5]

A smooth morphism has geometrically regular fibres, and an etale morphism is smooth (Étale morphism of schemes).

Proof

technique · direct. Everything is reduced to the lifting equivalence and the classification of finite etale algebras over the residue field

1.1F1algebra

Write $Y = Spec A$ with $A$ a finite etale $R$ -algebra. By [F1] the reduction functor $A \mapsto A \otimes_{R} k = A / m A$ is an equivalence from finite etale $R$ -algebras to finite etale $k$ -algebras. An equivalence is fully faithful, so it induces bijections ${Hom}_{R -alg} (A, R) ⟶ {Hom}_{k -alg} (A \otimes_{R} k, k),$ natural in $A$ ; under the anti-equivalence of affine schemes these are the maps $Y (R) \to Y (k)$ given by reduction. Hence $Y (R) \to Y (k)$ is a bijection.

1.2F2F3F4F5givenalgebra

Assume now that $k$ has no nontrivial finite separable extension, and let $E$ be a finite etale $k$ -algebra. As a finite-dimensional commutative $k$ -algebra, $E$ is Artinian, so $E ≅ \prod_{i} E_{i}$ with each $E_{i}$ local Artinian by [F4]. Each $E_{i}$ is a direct factor of $E$ , hence finite etale over $k$ ; being etale over the field $k$ it is smooth of relative dimension zero, so its only fibre is geometrically regular and in particular regular by [F5]. A regular local ring is a domain, and a local Artinian domain is a field by [F4], so $E_{i}$ is a field; as a finite etale field extension of $k$ it is finite separable over $k$ by [F3], hence equals $k$ . Therefore $E ≅ k^{d}$ for $d = \dim_{k} E$ , and every finite etale $k$ -algebra is a product of copies of $k$ .

2.1F1F2F3F4step 1.1step 1.2algebra∎

Let $A$ be a finite etale $R$ -algebra of rank $d$ ; by [F2] its rank equals the $k$ -dimension of $A \otimes_{R} k$ , which is a finite etale $k$ -algebra, so $A \otimes_{R} k ≅ k^{d}$ by step 1.2. Since reduction is an equivalence by [F1], it is essentially surjective and reflects isomorphisms, so $A ≅ R^{d}$ ; consequently $Y = Spec A$ is the disjoint union of $d$ copies of $Spec R$ . The complete Noetherian local hypotheses are exactly those stated, and AC is available for both the lifting equivalence [F1] and the classification suppliers [F2]-[F4].

DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passjudge pass (gpt-6.1-sol)Open item page →

S-dense open subschemes and S-rational maps

Definition

Let $S$ be a locally Noetherian scheme (Locally Noetherian and Noetherian schemes) and let $X$ and $Y$ be smooth $S$ -schemes (Smooth morphism of schemes). An open subscheme $U \subseteq X$ (Open immersions of schemes) is $S$ -dense if for every $s \in S$ the fibre $U_{s} = U \times_{S} Spec k (s)$ is Zariski dense in the fibre $X_{s} = X \times_{S} Spec k (s)$ (Scheme-theoretic fibre). Fiberwise, $(U \cap V)_{s} = U_{s} \cap V_{s}$ and the intersection of two dense open subsets of a topological space is dense; hence finite intersections of $S$ -dense open subschemes of $X$ are again $S$ -dense in $X$ . Similarly, if $U$ is $S$ -dense and open in $X$ and $V \subseteq X$ is open, then $U \cap V$ is $S$ -dense in $V$ , since $(U \cap V)_{s} = U_{s} \cap V_{s}$ is dense in $V_{s}$ .

An $S$ -rational map $u : X ⇢ Y$ is an equivalence class of $S$ -morphisms $U \to Y$ defined on $S$ -dense open subschemes $U \subseteq X$ , where two such morphisms $U \to Y$ and $U^{'} \to Y$ are equivalent if they coincide on an $S$ -dense open subscheme of $U \cap U^{'}$ . We say $u$ is defined at a point $x \in X$ if some representative is defined on an open subscheme containing $x$ . The union of the domains of all representatives is an $S$ -dense open subscheme $dom (u)$ , the domain of definition of $u$ . When $Y$ is separated the representatives agree on their intersections and glue to a morphism on $dom (u)$ ; without separatedness such a global representative need not exist. This is the relative version of Rational maps of integral finite-type schemes.

Base change. The notions $S$ -dense and $S$ -rational are preserved by arbitrary base change $S^{'} \to S$ . The domain of definition is compatible with flat base change in a sharp sense: if $X$ and $Y$ are smooth of finite type over $S$ and $Y$ is separated over $S$ , if $u : X ⇢ Y$ is an $S$ -rational map and $S^{'} \to S$ is flat, then the base-changed $S^{'}$ -rational map $u_{S^{'}}$ satisfies $dom (u_{S^{'}}) = dom (u) \times_{S} S^{'}$ (BLR 2.5/6, Proposition 6). Flatness is essential: over $S = Spec Z$ , the $S$ -rational map $A_{S}^{1} ⇢ A_{S}^{1}$ given by $2 / T$ on the $S$ -dense open $D (T)$ has domain exactly $D (T)$ , since $2 / T$ is not regular at any prime containing $T$ . After base change to $Spec F_{2}$ it is the zero rational map, which extends over the whole affine line. Thus the domain of definition does not commute with this non-flat base change.

A collection of fibrewise rational maps with informal specialization compatibility is not used on this page as an equivalent definition of an $S$ -rational map: an actual representative on an $S$ -dense open subscheme is required, and all extension arguments below produce such representatives.

LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

A normal Noetherian domain is the intersection of its height-one localizations

Statement

Assume AC. Let $A$ be a Noetherian normal domain (normal noetherian ring) with fraction field $K$ , so that $A$ is integrally closed in $K$ . Then, inside $K$ , $A = ⋂_{ht p = 1} A_{p},$ the intersection running over all height-one prime ideals $p$ of $A$ . Equivalently, a rational function which is regular at every height-one point of $Spec A$ is regular.

Facts & Assumptions

Given: AC, a Noetherian normal domain $A$ with fraction field $K$ , and an element $z = b / a \in K$ with $a, b \in A$ , $a \neq 0$ .

[F1]

A Noetherian ring is normal when every prime localization is an integrally closed domain; for a domain this means integrally closed in its fraction field (normal noetherian ring). A normal Noetherian domain satisfies Serre's condition $(S_{2})$ : $depth A_{p} \geq \min (2, ht p)$ for every prime $p$ (normal domain implies s two, assuming AC).

[F2]

A nonzero module over a Noetherian ring has an associated prime (A nonzero module over a Noetherian ring has an associated prime, assuming DC, hence in particular under AC); a prime $p$ is associated to $M$ exactly when $p = Ann (m)$ for some $m \in M$ , equivalently when $A / p$ embeds in $M$ (Associated primes are exactly primes of embedded cyclic residue modules).

[F3]

For a Noetherian local ring $(R, m)$ and a nonzero finite module $M$ , $depth (M) = 0$ if and only if $m \in Ass (M)$ (The local depth-zero associated-prime criterion, assuming AC).

[F4]

If $R$ is Noetherian, $M$ finite, $I \subseteq J (R)$ and $x \in I$ is $M$ -regular, then ${depth}_{I} (M / x M) = {depth}_{I} (M) - 1$ (Depth drops by one after quotienting by a regular element, assuming AC).

[F5]

A height-one prime localization of a Noetherian integrally closed domain is a discrete valuation ring (Height-one localizations of normal Noetherian domains are DVRs).

Proof

technique · direct, by contraposition of the nontrivial inclusion

1.1givenalgebra

The inclusion $A \subseteq ⋂_{ht p = 1} A_{p}$ holds because every $A_{p}$ contains $A$ , compatibly with the common fraction field $K$ ; all rings involved are subrings of $K$ .

2.1F2step 1.1algebra

Suppose $z = b / a \notin A$ . Then $b \notin a A$ , so the class $\overset{ˉ}{b}$ of $b$ in the finite nonzero $A$ -module $A / a A$ generates a nonzero cyclic submodule $C = A \overset{ˉ}{b}$ . By [F2] $C$ has an associated prime $p$ , say $p = Ann (c \overset{ˉ}{b})$ for some $c \in A$ ; then $0 \neq a \in p$ because $a$ annihilates $A / a A$ .

3.1F2F3step 2.1algebra

The localized module $C_{p}$ is nonzero, since $Ann (c \overset{ˉ}{b}) = p$ : if $s \in A ∖ p$ satisfied $s (c \overset{ˉ}{b}) = 0$ , then $s \in p$ , a contradiction. As $C_{p} \subseteq (A / a A)_{p} = A_{p} / a A_{p}$ and the annihilator of the image of $c \overset{ˉ}{b}$ in this localization is $p A_{p}$ , the associated-prime depth criterion [F3] gives ${depth}_{A_{p}} (A_{p} / a A_{p}) = 0$ .

4.1F4step 3.1algebra

Since $A$ is a domain and $a \neq 0$ , the element $a$ is $A_{p}$ -regular and lies in the maximal ideal $p A_{p} \subseteq J (A_{p})$ ; the regular-element depth formula [F4] applied to $M = A_{p}$ gives $depth (A_{p} / a A_{p}) = depth (A_{p}) - 1$ , so $depth A_{p} = 1$ .

5.1F1F5step 2.1step 4.1algebra

By the $(S_{2})$ condition of [F1], $1 = depth A_{p} \geq \min (2, ht p)$ , so $ht p \leq 1$ ; since $0 \neq a \in p$ we also have $ht p \geq 1$ , hence $ht p = 1$ , and $A_{p}$ is a discrete valuation ring by [F5].

6.1F2step 2.1step 5.1algebra∎

Finally $z \notin A_{p}$ . Indeed, for the annihilator $Ann (\overset{ˉ}{b}) \subseteq A$ of $\overset{ˉ}{b} \in A / a A$ one has $Ann (\overset{ˉ}{b}) \subseteq p$ , since $s \overset{ˉ}{b} = 0$ implies $s (c \overset{ˉ}{b}) = c (s \overset{ˉ}{b}) = 0$ and hence $s \in Ann (c \overset{ˉ}{b}) = p$ . If $b / a \in A_{p}$ , write $b = a u$ with $u = r / s$ , $r \in A$ , $s \in A ∖ p$ ; then $s b = a r \in a A$ , so $s \in Ann (\overset{ˉ}{b}) \subseteq p$ , contradicting $s \notin p$ . Thus every $z \in K$ outside $A$ lies outside $A_{p}$ for some height-one prime $p$ , and with step 1.1 the intersection equals $A$ .

The last step also yields the standard Hartogs form: an element of $K$ contained in $A_{p}$ for every height-one prime $p$ lies in $A$ , so on a normal Noetherian scheme a rational function regular in codimension one is regular.

DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passOpen item page →

Neron models, the Neron mapping property and weak Neron models

Definition

Let $S$ be a Dedekind scheme (Dedekind domains) with function field $K$ , and let $X_{K}$ be a smooth separated $K$ -scheme of finite type. For an $S$ -scheme $X$ , its generic fibre is $X_{K} = X \times_{S} Spec K$ , the fibre of $X \to S$ over the generic point of $S$ (Scheme-theoretic fibre); it is a $K$ -scheme. An $S$ -model of $X_{K}$ is an $S$ -scheme $X$ together with a specified isomorphism $X_{K} ≅ X \times_{S} Spec K$ of $K$ -schemes.

A Neron model of $X_{K}$ over $S$ is an $S$ -model $X \to S$ which is smooth (Smooth morphism of schemes), separated (Separated morphism of schemes), of finite type (Locally finite type and finite type morphisms), and which satisfies the Neron mapping property: for every smooth $S$ -scheme $Y$ and every $K$ -morphism $u_{K} : Y_{K} \to X_{K}$ there is a unique $S$ -morphism $u : Y \to X$ extending $u_{K}$ , that is, with $u \times_{S} Spec K = u_{K}$ under the specified identifications.

Equivalently, $X$ represents the functor $Y \mapsto {Hom}_{K} (Y_{K}, X_{K})$ from smooth $S$ -schemes to sets, so by the Yoneda lemma a Neron model of $X_{K}$ is determined up to a unique isomorphism: applying the property to $Y = X$ and to its identity $K$ -morphism shows that any two Neron models of $X_{K}$ admit a unique $S$ -isomorphism over $X_{K}$ .

Local nature. For a closed point $s \in S$ the local ring $O_{S, s}$ is a discrete valuation ring with fraction field $K$ , and $X \times_{S} Spec O_{S, s}$ is a Neron model of $X_{K}$ over the local Dedekind scheme $Spec O_{S, s}$ whenever $X$ is a Neron model of $X_{K}$ over $S$ ; conversely, an $S$ -model of finite type over $S$ is a Neron model if each of its localizations at closed points is. The finite-type hypothesis is essential for this converse. Thus the notion is local on $S$ .

Weak Neron models. A scheme $X$ over the Dedekind scheme $S$ satisfies the extension property for etale points at a closed point $s \in S$ if for each etale local $O_{S, s}$ -algebra $R^{'}$ (a local ring with a local homomorphism $O_{S, s} \to R^{'}$ that is etale, Étale morphism of schemes), with fraction field $K^{'}$ , the canonical map $X (R^{'}) \to X_{K} (K^{'})$ is surjective. A weak Neron model of $X_{K}$ is a smooth separated finite-type $S$ -model $X$ of $X_{K}$ satisfying the extension property for etale points at every closed point of $S$ . When $X$ is separated over $S$ the displayed map is injective by the valuative criterion of separatedness, so the extension property is then a bijection.

The Neron mapping property applied to $Y = X$ shows that a Neron model of $X_{K}$ is unique up to a unique isomorphism inducing the specified identity on $X_{K}$ and, applied to etale $S$ -schemes $Y$ , shows that a Neron model is in particular a weak Neron model. This definition asserts no existence statement: it describes what it means for a model to be a Neron model, and every existence claim on this page is a theorem with its own hypotheses. The weak Neron property is not asserted here to characterize Neron models.

TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6.1-sol)Open item page →

The chord-tangent group law on a smooth short Weierstrass cubic

Statement

Assume AC, inherited from Riemann-Roch and scheme-theoretic descent. Let $k$ be a field of characteristic different from $2, 3$ , let $a, b \in k$ with $4 a^{3} + 27 b^{2} \neq 0$ , and let $C = V_{+} (Y^{2} Z - X^{3} - a X Z^{2} - b Z^{3}) \subseteq P_{k}^{2}$ be the short Weierstrass cubic with origin $O = [0 : 1 : 0]$ .

Then $C$ is a smooth projective geometrically integral curve of genus one over $k$ , and the chord-tangent law makes it an abelian variety over $k$ : for $P = (x_{1}, y_{1})$ and $Q = (x_{2}, y_{2})$ in the affine chart $Z \neq 0$ with $P + Q \neq O$ , the slope is $λ = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{x_{1}^{2} + x_{1} x_{2} + x_{2}^{2} + a}{y_{1} + y_{2}}$ whenever the chosen denominator is nonzero, and the sum is $x (P + Q) = λ^{2} - x_{1} - x_{2}, y (P + Q) = λ (x_{1} - x (P + Q)) - y_{1},$ while $P + Q = O$ for inverse pairs and for a doubled two-torsion point; the inverse is $[X : Y : Z] \mapsto [X : - Y : Z]$ . The law is a morphism $C \times_{k} C \to C$ , and all group identities hold as morphisms.

Facts & Assumptions

Given: AC, a field $k$ with $char k \neq 2, 3$ , elements $a, b \in k$ with $4 a^{3} + 27 b^{2} \neq 0$ , and the cubic $C = V_{+} (Y^{2} Z - X^{3} - a X Z^{2} - b Z^{3})$ with origin $O = [0 : 1 : 0]$ .

[F1]

A curve over $k$ is a geometrically integral, separated, finite-type $k$ -scheme of dimension one; $C$ is a closed subscheme of $P_{k}^{2}$ , which is proper over $k$ , and the Jacobian criterion detects smoothness geometrically (Curves over a field, Finite-dimensional projective space is proper over every base, Relative Jacobian criterion with its presentation hypothesis, projective space points, Scheme-theoretic fibre).

[F2]

A smooth plane curve of degree $d$ has genus $(d - 1) (d - 2) / 2$ , so a smooth plane cubic has genus one (The genus of a smooth plane curve in terms of its degree). Two plane curves of degrees $d, e$ without common component meet in a divisor of degree $d e$ , weighted by local length and residue degree (Algebraic Bezout formula as a sum of local scheme lengths, assuming AC).

[F3]

On a smooth proper geometrically integral genus-one curve, $ω_{C} ≅ O_{C}$ and $h^{0} (ω_{C}) = 1$ ; Riemann-Roch reads $l (D) - l (K_{C} - D) = \deg D$ (The canonical bundle of a genus-one curve is trivial, The full Riemann-Roch theorem for divisors on a smooth proper curve, both assuming AC). The degree is a homomorphism $\deg ⁣ : Pic (C) \to Z$ with kernel ${Pic}^{0} (C)$ (Picard group of a scheme, The degree of a divisor descends to the Picard group of a normal proper curve, assuming AC). A genus-one curve with a rational point has a degree-three very ample line bundle embedding it as a plane cubic (A genus-one curve with a rational point embeds as a plane cubic).

[F4]

A rational map from a smooth curve over $k$ to a proper $k$ -scheme extends to a morphism; two morphisms from a reduced source that agree on a dense open are equal (Rational maps from a smooth curve to a proper scheme are morphisms, Agreement on a schematically dense open, both assuming AC).

[F5]

Morphisms satisfying an fppf descent datum descend; morphisms between finitely presented schemes over a filtered colimit of fields descend to a finite stage; algebraic closures exist (Scheme morphisms satisfy fppf descent, Finite-stage descent of finitely presented schemes and their morphisms, Assuming Choice, every field has an algebraic closure, assuming AC). The group scheme conventions are Abelian varieties over a field.

Proof

technique · direct: identify the chord-tangent operation with addition of divisor classes, then show it is algebraic and descends

1.1F1F2givenalgebra

The cubic $C$ is smooth over $k$ : on the affine chart $Z = 1$ a common zero of $\partial F / \partial x = - 3 x^{2} - a$ and $\partial F / \partial y = 2 y$ would force $y = 0$ , $3 x^{2} = - a$ and $b - 2 x^{3} = 0$ , hence $4 a^{3} + 27 b^{2} = 0$ , and in the chart $Y = 1$ the gradient at $O$ is nonzero because $\partial (Z - X^{3} - a X Z^{2} - b Z^{3}) / \partial Z = 1$ at $O$ ; the Jacobian criterion is compatible with field extension, so $C$ is smooth over every extension of $k$ . If $C_{\overset{ˉ}{k}}$ were reducible, its components would be plane curves of positive degrees summing to $3$ , and by [F2] they would meet in a nonempty divisor, at whose points $C$ would not be regular, a contradiction; hence $C$ is geometrically integral and, being a closed subscheme of $P_{k}^{2}$ , proper of dimension one over $k$ . Its genus is one by [F2], so $C$ is a curve in the sense of [F1] of genus one.

2.1F3step 1.1algebra

Since $O \in C (k)$ , the class $O_{C} (3 O)$ has degree $3$ , and the assignment $φ (P) = [O_{C} (P - O)]$ defines a map $C (\overset{ˉ}{k}) \to {Pic}^{0} (C_{\overset{ˉ}{k}})$ : the difference of two degree-one divisors has degree zero. It is bijective. Indeed, for $[L] \in {Pic}^{0} (C_{\overset{ˉ}{k}})$ the sheaf $L \otimes O_{C} (O)$ has degree one and $l = \deg = 1$ by Riemann-Roch and triviality of the canonical bundle in [F3], so it admits a nonzero section whose divisor $P$ is effective of degree one; then $[O (P - O)] = [L]$ . Uniqueness holds because $l (O (P)) = 1$ , so the effective divisor of degree one in a degree-one class is unique.

3.1F3step 2.1algebra

Let $P, Q \in C (\overset{ˉ}{k})$ and let $ℓ$ be the line through $P$ and $Q$ , tangent at $P$ when $P = Q$ ; write $P + Q + R$ for the divisor of the corresponding hyperplane section, which has degree $3$ by [F2]. The restriction of $O (1)$ to $C$ is isomorphic to $O_{C} (3 O)$ : the coordinate $Z$ restricts to a section whose divisor is $3 O$ , since on the chart $Y = 1$ the equation of $C$ is $Z = X^{3} + a X Z^{2} + b Z^{3}$ , so the intersection with the line $Z = 0$ is the point $X = Z = 0$ with multiplicity three. Hence $P + Q + R \sim 3 O$ . For the vertical line through $R$ the intersection divisor is $R + (- R) + O$ , so $R + (- R) \sim 2 O$ . Combining, $P + Q - (- R) \sim O$ , that is, in ${Pic}^{0}$ , $φ (- R) = φ (P) + φ (Q)$ : the assignment of step 2.1 converts the chord-tangent operation $P + Q : = - R$ into addition in the abelian group ${Pic}^{0} (C_{\overset{ˉ}{k}})$ . Consequently the operation is commutative and associative, has identity $O$ , and inverse $- ⁣ P = [X : - Y : Z] (P)$ ; and in the affine chart the standard substitution of the line $y = λ x + ν$ in $y^{2} = x^{3} + a x + b$ gives the displayed formulas, with $λ$ as in the statement when the chosen denominator is nonzero.

4.1F1F2step 1.1step 3.1algebra

The operation is algebraic. On $(C ∖ {O}) \times (C ∖ {O})$ with affine coordinates $(x_{i}, y_{i})$ define $(N, D) = (y_{2} - y_{1}, x_{2} - x_{1})$ on the first open and $(N, D) = (x_{1}^{2} + x_{1} x_{2} + x_{2}^{2} + a, y_{1} + y_{2})$ on the second, and consider the morphism $(P, Q) \mapsto [D (N^{2} - (x_{1} + x_{2}) D^{2}) : N ((2 x_{1} + x_{2}) D^{2} - N^{2}) - y_{1} D^{3} : D^{3}]$ . On $D \neq 0$ this is the sum computed in step 3.1 with $λ = N / D$ , and on $D = 0$ , $N \neq 0$ its value is $O$ , which is the sum of the inverse pair $P, Q$ . The two opens cover the affine square: if the first pair $(N, D)$ vanishes then $x_{1} = x_{2}$ and $y_{1} = y_{2}$ , i.e. $P = Q$ , and then the second pair is $(3 x_{1}^{2} + a, 2 y_{1})$ , which cannot vanish at a point of the smooth curve $C$ by step 1.1. Hence the sum is a morphism on $(C ∖ {O})^{2}$ ; the extension at pairs involving $O$ is established next.

5.1F4step 3.1step 4.1algebra

Over $\overset{ˉ}{k}$ the law extends to the full product and its group identities hold: for each $R \in C (\overset{ˉ}{k})$ the translation $T_{R}$ is a rational map from the smooth curve $C_{\overset{ˉ}{k}}$ to the proper scheme $C_{\overset{ˉ}{k}}$ , hence a morphism by [F4]; $T_{R}$ and $T_{- R}$ are mutually inverse on a dense open and hence everywhere by [F4]; for arbitrary $(P, Q)$ and $R$ avoiding $- P$ and $Q$ the expression $(P + R) + (Q - R)$ is defined and regular near $(P, Q)$ , these local morphisms agree on dense opens and hence glue to a morphism $C_{\overset{ˉ}{k}} \times C_{\overset{ˉ}{k}} \to C_{\overset{ˉ}{k}}$ extending the law of step 4.1. Since the group identities are identities of morphisms between reduced schemes over $\overset{ˉ}{k}$ and hold on the dense set of $\overset{ˉ}{k}$ -points described in step 3.1, they hold everywhere by [F4]; the inverse $[X : - Y : Z]$ is a regular involution.

6.1F4F5step 4.1step 5.1algebra∎

Since $C$ and $C \times_{k} C$ are finitely presented, [F5] descends the morphism of step 5.1 to some finite extension $L / k$ inside $\overset{ˉ}{k}$ . Its restriction to $U = (C ∖ {O})^{2}$ is the $k$ -defined morphism of step 4.1: this equality can be checked after the faithfully flat extension $\overset{ˉ}{k} / L$ . The two pullbacks over $L \otimes_{k} L$ therefore agree on $U_{L \otimes_{k} L}$ . This open is schematically dense in $(C \times_{k} C)_{L \otimes_{k} L}$ : on affine charts restriction to the dense open is injective before base change, and a finite principal-open cover computes its sections by a finite equalizer; tensoring over the field $k$ preserves these injections and equalizers. Thus separatedness and [F4] make the pullbacks equal even when $L \otimes_{k} L$ is nonreduced. The finite faithfully flat extension $L / k$ is an fppf cover, so [F5] descends the law to $k$ . The unit $O$ and inversion are already defined over $k$ , and the group identities hold after the faithfully flat extension to $\overset{ˉ}{k}$ , hence over $k$ . The smooth proper geometrically integral curve $C$ with this law is an abelian variety.

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Prime-to-characteristic torsion bound for affine commutative groups

Statement

Assume AC and DC (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). Let $k$ be an algebraically closed field and let $N$ be a smooth, connected, commutative affine group scheme of finite type over $k$ ; put $a = \dim N$ . For every prime $ℓ \neq char k$ and every integer $ν \geq 1$ , $∣ N [ℓ^{ν}] (k) ∣ \leq ℓ^{ν a},$ where $N [ℓ^{ν}] = \ker (ℓ^{ν} ⁣ : N \to N)$ is the $ℓ^{ν}$ -torsion subgroup scheme and $N [ℓ^{ν}] (k)$ is its group of $k$ -points.

Facts & Assumptions

Given: AC and DC, an algebraically closed field $k$ , a smooth connected commutative affine finite-type $k$ -group $N$ with $a = \dim N$ , a prime $ℓ \neq char k$ and an integer $ν \geq 1$ .

[F1]

The right regular representation of an affine finite-type group scheme over an arbitrary field contains a finite-dimensional subrepresentation $V$ with $G \to GL (V)$ a closed immersion, allowing nonreduced $G$ (Affine finite-type group schemes have faithful finite-dimensional representations).

[F2]

A smooth connected finite-type $k$ -group scheme is geometrically integral; over the algebraically closed field $k$ this says $k [N]$ is a domain (Connected finite-type groups are geometrically connected, which assumes AC).

[F3]

Over an algebraically closed field, the strong Nullstellensatz says that the ideal of polynomials vanishing on the zero locus of an ideal in a polynomial ring is its radical (Strong Nullstellensatz: I(V(I)) equals the radical of I, which assumes AC).

[F4]

For an abelian group $M$ the group algebra $k [M]$ has $k$ -basis the distinct group-like elements $e_{m}$ , with $Δ (e_{m}) = e_{m} \otimes e_{m}$ , and $D_{k} (M) (R) = Hom (M, R^{\times})$ for every $k$ -algebra $R$ (Diagonalizable groups and their character modules, Split diagonalizable groups are dual to abelian groups).

[F5]

For a finite-type $k$ -domain $A$ with fraction field $K$ , $\dim A = {trdeg}_{k} K$ (Affine-domain dimension equals transcendence degree).

Proof

technique · direct. The proof triangulates the matrices of $k$-points, identifies the diagonal quotient as a diagonalizable group with torsion-free character group of rank at most $a$, and compares torsion on the two sides

1.1F1givenconstruct

By [F1] fix a closed immersion $j : N \to {GL}_{m}$ of $k$ -group schemes, and for $g \in N (k)$ let $M (g) \in {GL}_{m} (k)$ be its matrix. Since $j$ is a group homomorphism and $N$ is commutative, $M (g) M (h) = M (g h) = M (h) M (g)$ , so the $k$ -span $A \subseteq {End}_{k} (k^{m})$ of the set ${M (g) : g \in N (k)}$ is a finite-dimensional commutative subalgebra. Let $W \subseteq k^{m}$ be a nonzero $A$ -stable subspace of minimal dimension. If some $A$ -element has an eigenspace $C$ inside $W$ with $0 \neq C \neq W$ , then $C$ is a smaller nonzero $A$ -stable subspace, a contradiction; if every element of $A$ acts as a scalar on $W$ , then every line in $W$ is $A$ -stable; hence in either case $\dim_{k} W = 1$ and $W$ is a common eigenvector. Applying this argument to $k^{m}$ and then to the successive quotients by the lines produced yields a filtration $0 = V_{m} \subset V_{m - 1} \subset \dots \subset V_{0} = k^{m}$ with $M (g) V_{i} \subseteq V_{i}$ for all $i$ and all $g \in N (k)$ ; in a basis adapted to this filtration every $M (g)$ is upper triangular.

2.1F2F3step 1.1algebra

Put $P = k [x_{i j}]$ , let $J = \ker (P \to k [N])$ be the kernel of the map sending each matrix coordinate to its pullback under $j$ , and write $δ = \det (x_{i j})$ . The map $P_{δ} = k [x_{i j}, δ^{- 1}] \to k [N]$ is surjective because $j$ is a closed immersion into ${GL}_{m}$ , and its kernel is $J_{δ}$ . For every $i > j$ , $x_{i j}$ vanishes at every common zero of $J$ and $z δ - 1$ in the polynomial ring $P [z]$ : such a zero is an invertible matrix satisfying the equations of the closed subscheme $N \subseteq {GL}_{m}$ , hence is a $k$ -point of $N$ and is upper triangular by step 1.1. Applying [F3] in $P [z]$ gives $x_{i j} \in \sqrt{(J, z δ - 1)}$ . After quotienting by $z δ - 1$ and identifying $P [z] / (z δ - 1) ≅ P_{δ}$ , this says $x_{i j} \in \sqrt{J_{δ}}$ . Since $P_{δ} / J_{δ} ≅ k [N]$ is a domain by [F2], $J_{δ}$ is radical and therefore $x_{i j} \in J_{δ}$ . Thus the coordinate functions below the diagonal vanish scheme-theoretically on $N$ , so $j$ factors through the closed subgroup scheme $B \subseteq {GL}_{m}$ of upper triangular matrices. In particular the diagonal characters $χ_{i} : = u_{i i}$ (the images in $k [N]$ of the diagonal coordinate functions) are units and satisfy $Δ (χ_{i}) = χ_{i} \otimes χ_{i}$ in the Hopf algebra $k [N]$ .

3.1F4step 2.1algebra

Let $R \subseteq k [N]$ be the subalgebra generated by $χ_{1}^{\pm 1}, \dots, χ_{m}^{\pm 1}$ . It is a Hopf subalgebra because the $χ_{i}$ are group-like units. Its group-like elements are the monomials $χ^{(n)} = \prod_{i} χ_{i}^{n_{i}}$ for $n \in Z^{m}$ , and distinct group-like elements of a Hopf algebra over a field are linearly independent: a shortest nontrivial relation $\sum_{i \in S} c_{i} h_{i} = 0$ with distinct $h_{i}$ and all $c_{i} \neq 0$ becomes, after applying $Δ$ and subtracting the tensor product of the relation with $h_{s}$ , the relation $\sum_{i \in S ∖ {s}} c_{i} h_{i} \otimes (h_{i} - h_{s}) = 0$ ; the $h_{i}$ for $i \neq s$ are linearly independent by minimality of $S$ , so $h_{i} = h_{s}$ for all $i$ , a contradiction. Hence $R = k [M]$ where $M = Z^{m} / L$ is the quotient of $Z^{m}$ by the relations $L = {n : χ^{(n)} = 1}$ , with $k$ -basis the distinct $χ^{(n)}$ ; by [F4] this is the coordinate Hopf algebra of the diagonalizable group $D_{k} (M)$ , and $N \to D_{k} (M)$ is the morphism dual to the inclusion $R \subseteq k [N]$ .

4.1F2F4F5step 3.1algebra

The ring $k [M]$ is a domain, being a subalgebra of the domain $k [N]$ by [F2]. If $M$ had a nonzero element $n$ of finite order $d > 1$ , then $(χ^{(n)} - 1) (χ^{(n)^{d - 1}} + \dots + χ^{(n)} + 1) = χ^{(n)^{d}} - 1 = χ^{(d n)} - 1 = 0$ in $k [M]$ , and both factors are nonzero because distinct group-like elements are linearly independent by step 3.1 and the second factor is a sum of distinct group-likes with coefficient $1$ . This contradicts that $k [N]$ is a domain, so $M$ is torsion-free; as a finitely generated torsion-free abelian group, $M ≅ Z^{t}$ for some $t \geq 0$ . Since $R = k [M]$ is a finite-type $k$ -domain contained in $k [N]$ , its fraction field embeds in $Frac (k [N]) = k (N)$ , whence $t = \dim k [M] = {trdeg}_{k} Frac (k [M]) \leq {trdeg}_{k} k (N) = \dim k [N] = a$ by [F5].

5.1givenstep 3.1step 4.1algebra

Let $φ : N (k) \to D_{k} (M) (k) = Hom (M, k^{\times})$ be the homomorphism induced by $N \to D_{k} (M)$ . A point $g \in N (k)$ lies in $\ker φ$ precisely when $χ_{i} (g) = 1$ for all $i$ , that is, precisely when its matrix $M (g)$ is upper unitriangular; write $M (g) = I + U$ with $U$ strictly upper triangular, so $U^{m} = 0$ . If $char k = 0$ and $g$ has finite order $d$ , then $(M (g) - I)^{m} = 0$ and $M (g)^{d} = I$ , so the minimal polynomial of $M (g)$ divides both $(X - 1)^{m}$ and $X^{d} - 1$ ; in characteristic zero $X^{d} - 1$ is squarefree, so the gcd is $X - 1$ and $M (g) = I$ . If $char k = p > 0$ and $p^{e} \geq m$ , then $M (g)^{p^{e}} = I + U^{p^{e}} = I$ , so $M (g)$ has $p$ -power order. Hence an element of $N (k)$ of order dividing $ℓ^{ν}$ with $ℓ \neq p$ that lies in $\ker φ$ equals the identity: it has order dividing both $ℓ^{ν}$ and a $p$ -power, hence order $1$ . Consequently the restriction $N [ℓ^{ν}] (k) \to D_{k} (M) (k) [ℓ^{ν}]$ is injective.

6.1F4step 4.1step 5.1algebra∎

Since $M ≅ Z^{t}$ , evaluation gives $D_{k} (M) (k) = Hom (Z^{t}, k^{\times}) ≅ (k^{\times})^{t}$ , whose $ℓ^{ν}$ -torsion is $Hom (Z^{t}, μ_{ℓ^{ν}} (k)) ≅ (Z / ℓ^{ν} Z)^{t}$ because $k$ is algebraically closed and $ℓ \neq char k$ , of cardinal $ℓ^{ν t}$ . By step 5.1, $∣ N [ℓ^{ν}] (k) ∣ \leq ∣ D_{k} (M) (k) [ℓ^{ν}] ∣ = ℓ^{ν t}$ , and $t \leq a$ by step 4.1, so $∣ N [ℓ^{ν}] (k) ∣ \leq ℓ^{ν a}$ . AC is used through [F2] and [F3]; DC is inherited from the faithful-representation supplier chain [F1] as declared.

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Strict henselization of a DVR and smooth sections

Statement

Assume AC and DC. Let $R$ be a discrete valuation ring with fraction field $K$ , residue field $k$ and a fixed uniformizer $π$ , and fix a separable closure $k^{s}$ of $k$ .

There exists a strictly henselian discrete valuation ring $R^{s h}$ , the strict henselization of $R$ , which is the filtered colimit of the pointed local etale $R$ -algebras with residue embeddings into $k^{s}$ . It is faithfully flat over $R$ , has residue field $k^{s}$ , has uniformizer $π$ , and is a filtered colimit of local etale $R$ -algebras.

Let $Λ$ be a strictly henselian local ring with separably closed residue field $κ$ (for instance $Λ = R^{s h}$ ), and let $V$ be a smooth $Λ$ -scheme. Every $κ$ -point of the special fibre $V_{κ}$ lifts to a $Λ$ -section of $V$ ; the set of specializations of sections of $V$ is dense in $V_{κ}$ . For the final assertion, assume additionally that $Λ$ is a strictly henselian DVR with fraction field $F$ and uniformizer $π$ . If $V$ is smooth, integral and of finite type over $Λ$ of pure relative dimension $d$ with nonempty special fibre $V_{κ}$ , and $U \subseteq V_{F}$ is a dense open subscheme of its generic fibre, then some $Λ$ -section of $V$ has generic point in $U$ ; for $Λ = R^{s h}$ this is the specialization of BLR 5.3/7 used in the finite translate enlargement.

Facts & Assumptions

Given: AC and DC, a DVR $R$ with fraction field $K$ , residue field $k$ , uniformizer $π$ , and separable closure $k^{s}$ ; $R^{s h}$ , $Λ$ , $κ$ , $V$ and $U$ as in the statement.

[F1]

Henselian local rings are characterized by unique lifting of simple roots of monic polynomials (Henselian pairs and Henselian local rings, A local ring is Henselian exactly when simple residue roots lift uniquely). Together with the standard-etale charts [F2], this lifts a residual rational point of an etale neighbourhood uniquely.

[F2]

An etale morphism is locally standard etale: locally on source and target it is a localization of a monogenic presentation by a monic polynomial with invertible derivative (Étale morphisms are locally standard étale, which assumes AC).

[F3]

A locally finitely presented morphism is smooth at a point exactly where a standard smooth chart with a unit Jacobian minor exists; such charts are flat with geometrically regular fibres (Relative Jacobian criterion with its presentation hypothesis, which assumes AC).

[F4]

A nonzero Noetherian local ring of dimension one is regular if and only if it is a discrete valuation ring, and every DVR is a principal ideal domain (one dimensional regular local rings are dvrs, Every DVR is a PID).

[F5]

Over a principal ideal domain flatness is equivalent to torsion-freeness (Over a principal ideal domain flatness is equivalent to torsion-freeness); a flat local ring homomorphism whose closed fibre is nonzero is faithfully flat (A flat ring map is faithfully flat exactly when it detects proper ideals and is surjective on spectra).

[F6]

Every nonempty smooth finite-type scheme over a field has a closed point with finite separable residue field; over a separably closed field this is a rational point (A nonempty smooth scheme has a finite separable point, which assumes AC).

Proof

technique · direct. The strict henselization is built as a filtered union of pointed etale neighbourhoods, and sections are produced from standard smooth charts by the henselian lifting property

1.1F2F4givenconstruct

Consider the directed system of pairs $(R^{'}, α)$ , where $R^{'}$ is a local $R$ -algebra which is etale over $R$ and whose residue field embeds as an $R$ -subalgebra $k \to R^{'} / m_{R^{'}} ↪ k^{s}$ over those already chosen, with transition maps the local $R$ -algebra maps over $k^{s}$ . Each such $R^{'}$ is flat over $R$ because etale maps are flat; its residue field is a finite separable extension of $k$ , so $m_{R^{'}} = π R^{'}$ and $\dim R^{'} = 1$ ; being regular of dimension one it is a DVR with uniformizer $π$ by [F4]. The transition maps are injective local maps preserving $π$ . In the union $R^{s h} = colim R^{'}$ over the directed system, every nonzero element lies in a finite stage as $π^{n}$ times a unit, and every nonzero ideal has a least such exponent, so it is principal; hence $R^{s h}$ is a DVR with uniformizer $π$ and residue field $k^{s}$ , realized as a filtered colimit of local etale $R$ -algebras.

2.1F1F5step 1.1construct

The ring $R^{s h}$ embeds into a separable closure of $K$ and hence is $R$ -torsion-free, so it is $R$ -flat by [F5] and $R$ -faithfully flat because it is local with nonzero closed fibre. Its strict henselianity follows from the colimit description: an etale neighbourhood of a point with residual coefficients involves finitely many elements of $R^{s h}$ , hence is defined at a finite stage, and adjoining that pointed neighbourhood to the directed system exhibits its residual lift in the limit; this proves the henselian neighbourhood-lifting criterion of [F1] without assuming the individual stages are henselian.

3.1F1F3step 2.1construct

Let $V$ be smooth over $Λ$ and let $x \in V_{κ} (κ)$ . By [F3] choose a standard smooth chart $U = Spec C$ around $x$ with $C = (Λ [t_{1}, \dots, t_{n}] / (f_{1}, \dots, f_{m}))_{g}$ and a unit $m \times m$ Jacobian minor. Cutting the chart by the $n - m$ coordinate differences $t_{i} - \tilde{t_{i} (x)}$ with chosen lifts of the residue coordinates to $Λ$ that are not involved in that minor produces an etale $Λ$ -scheme through $x$ : the original $m$ equations and the $n - m$ coordinate differences have a unit $n \times n$ Jacobian minor. Its special fibre has the same $κ$ -point $x$ . The henselian lifting property of [F1] gives a $Λ$ -section of this etale neighbourhood, hence a section of $V$ through $x$ .

4.1F3F6step 3.1algebra

Every nonempty open of the smooth special fibre contains a $κ$ -rational point by [F6], since $κ$ is separably closed; step 3.1 lifts that point to a section of $V$ . Thus the specializations of sections meet every nonempty open and are dense in $V_{κ}$ . Empty special fibre makes the density assertion vacuous.

5.1F3F4F6step 3.1algebra∎

Now suppose $Λ$ is a strictly henselian DVR with fraction field $F$ , $V$ is smooth integral of finite type with nonempty special fibre, and $U \subset V_{F}$ is dense. Give $V_{F} ∖ U$ its reduced closed structure and let $Z$ be its schematic closure. Its ideal is saturated under multiplication by $π$ , since it contracts an ideal after inverting $π$ . At a generic point $ξ$ of a special-fibre component, the local ring $B = O_{V, ξ}$ has maximal ideal $(π)$ , because the smooth special fibre is reduced and its local ring there is a field. Since $V$ is integral and flat, $π$ is a nonzero nonunit; the principal ideal theorem (Krull's principal ideal theorem) gives $\dim B = 1$ , and $B$ is regular, hence a DVR by [F4]. The localized ideal of $Z$ is nonzero (the generic complement is proper in the integral $V$ ) and $π$ -saturated, so it is all of $B$ : every nonzero proper DVR ideal is $(π^{n})$ and fails saturation. Thus $Z$ misses every special-fibre generic point. A rational point of the nonempty smooth open $V_{κ} ∖ Z$ exists by [F6] and lifts to a section by step 3.1. Its generic point cannot lie in the closed $Z$ , since its specialization does not. This gives a section with generic point in $U$ .

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Finite Cartier duality, exactness and exponent

Statement

Assume AC and DC. Let $k$ be a field. A finite $k$ -scheme is an affine $k$ -scheme whose coordinate ring is a finite-dimensional $k$ -algebra; the rank of a finite $k$ -group scheme $H$ is $\dim_{k} O (H)$ .

Let $H$ be a finite commutative $k$ -group scheme with coordinate algebra $B = O (H)$ of rank $d$ . The vector-space dual $B^{*} = {Hom}_{k} (B, k)$ , with multiplication dual to $Δ_{B}$ and comultiplication dual to the multiplication of $B$ , is a commutative Hopf $k$ -algebra, and $H^{D} = Spec B^{*}$ is a finite commutative $k$ -group scheme of rank $d$ representing the functor $R ⟼ {Hom}_{R -groups} (H_{R}, G_{m, R})$ on $k$ -algebras: the $R$ -points of $H^{D}$ are the group-like elements of $B \otimes_{k} R$ , equivalently the all-test characters of $H_{R}$ . Evaluation and double vector-space duality give a natural Hopf isomorphism $H ≅ H^{D D}$ , and $H \mapsto H^{D}$ is functorial in $H$ .

Cartier duality is a contravariant additive equivalence of the category of finite commutative $k$ -group schemes with itself, and it preserves exactness with arrows reversed; a sequence of finite commutative $k$ -group schemes is exact if and only if its Cartier dual is. A finite commutative $H$ of rank $d$ is killed by $d$ : the endomorphism $[d]_{H}$ is zero. This holds for nonreduced $H$ and for $char k$ dividing $d$ .

In particular, for $n \geq 1$ there is an isomorphism $μ_{n}^{D} ≅ (Z / n)_{k}$ of finite commutative $k$ -group schemes.

Facts & Assumptions

Given: AC, DC, a field $k$ , a finite commutative $k$ -group scheme $H$ of rank $d$ , and an integer $n \geq 1$ .

[F1]

The coordinate algebra of an affine $k$ -group scheme is a commutative Hopf $k$ -algebra with comultiplication $Δ$ , counit $ϵ$ and antipode $S$ ; a group-like element is an element $a$ with $Δ (a) = a \otimes a$ and $ϵ (a) = 1$ (Coordinate Hopf algebras for multiplicative type).

[F2]

Affine $k$ -group schemes are contravariantly equivalent to commutative Hopf $k$ -algebras, a group character $G \to G_{m}$ corresponds precisely to a group-like element of its coordinate algebra, and these correspondences commute with field extension (The affine Hopf dictionary used for multiplicative type).

[F3]

The quotient $G / H$ of a separated finite-type $k$ -group scheme by a closed normal subgroup scheme is represented by a separated finite-type $k$ -group scheme, the projection is faithfully flat of finite presentation with scheme-theoretic kernel $H$ , and the quotient is universal for homomorphisms killing $H$ (Normal subgroup quotients of finite-type group schemes exist as fppf scheme quotients, which assumes AC).

[F4]

A homomorphism of separated finite-type $k$ -group schemes with trivial scheme-theoretic kernel is a closed immersion, and its scheme-theoretic image is a closed subgroup scheme (Finite-type algebraic group monomorphisms are closed immersions, which assumes AC).

[F5]

A commutative Artinian ring is the product of its localizations at its finitely many maximal ideals (An Artinian ring is canonically the finite product of its localizations at its maximal ideals); in particular a zero-dimensional finite-type $k$ -scheme is finite.

[F6]

A tensor product of free modules is free with the pairwise tensor basis, and the dual of a finite free module has the dual basis (The elementary tensors of two bases form the product basis of the tensor product).

[F7]

For a square matrix $A$ over a commutative ring, $A adj (A) = \det (A) I$ ; in particular multiplication by a unit of a finite free algebra has invertible determinant (For every positive-sized square matrix over a commutative ring, $A adj (A) = adj (A) A = \det (A) I$ ).

Proof

technique · direct. The dual Hopf algebra is constructed by transposing structure maps, the dual functor is identified by the character/group-like dictionary, and the exponent is obtained from a determinant argument, so no reducedness or separability hypothesis enters

1.1F1F2F6algebra

Let $B = O (H)$ and $B^{*} = {Hom}_{k} (B, k)$ . Define the comultiplication of $B^{*}$ as the transpose of the multiplication of $B$ , its multiplication as the transpose of $Δ_{B}$ , its unit as the transpose of $ϵ$ , its counit as the transpose of the unit of $B$ , and its antipode as the transpose of the antipode. Transposing the commutative diagrams that express coassociativity, the counit and antipode identities, commutativity of $B$ and cocommutativity of $Δ_{B}$ (the latter because $H$ is commutative) gives the corresponding identities for $B^{*}$ : finite-dimensional duality is an exact contravariant equivalence of finite-dimensional $k$ -vector spaces and carries commutative diagrams to commutative diagrams. Hence $B^{*}$ is a commutative Hopf $k$ -algebra with $\dim_{k} B^{*} = d$ , and $Spec B^{*}$ is an affine $k$ -group scheme of rank $d$ by [F1] and [F2].

2.1F1F2step 1.1algebra

For every $k$ -algebra $R$ , an $R$ -point of $Spec B^{*}$ , that is, a $k$ -algebra map $B^{*} \to R$ , corresponds by transpose to a group-like element of $B \otimes_{k} R$ : multiplicativity and unitality of the map are exactly the identities $Δ (χ) = χ \otimes χ$ and $ϵ (χ) = 1$ for the transpose $χ$ . By the character/group-like dictionary in [F2] these are exactly the $R$ -group homomorphisms $H_{R} \to G_{m, R}$ , and the correspondence is natural in $R$ . Therefore $H^{D} = Spec B^{*}$ represents the stated functor, and transposing a Hopf map $B \to B^{'}$ dualizes to a Hopf map $(B^{'})^{*} \to B^{*}$ , so $H \mapsto H^{D}$ is a functor.

3.1F1F2step 1.1step 2.1algebra

The evaluation map $B \to (B^{*})^{*}$ is an isomorphism of $k$ -vector spaces because $B$ is finite dimensional, and it is compatible with $Δ$ , the multiplication, the unit, the counit and the antipode, since both sides are obtained by transposing the structure maps twice; hence it is a Hopf isomorphism, and accordingly $H ≅ H^{D D}$ naturally. The same evaluation pairing gives the stated biduality.

4.1F6F7step 3.1algebra

Fix a $k$ -algebra $R$ , a point $h \in H (R)$ and a character $χ \in H^{D} (R)$ ; by step 2.1 the character $χ$ is a group-like unit of $B_{R} = B \otimes_{k} R$ . Translation by $h$ is the $R$ -automorphism $t_{h}$ of $H_{R}$ with inverse $t_{h^{- 1}}$ , so it induces an $R$ -algebra automorphism $t_{h}^{*}$ of $B_{R}$ , and multiplication by $χ$ induces an invertible $R$ -linear endomorphism $m_{χ}$ of the free $R$ -module $B_{R}$ of rank $d$ . Because $χ$ is a character, $m_{t_{h}^{*} χ} = t_{h}^{*} m_{χ} (t_{h}^{*})^{- 1}$ and $(t_{h}^{*} χ) (b) = χ (b h) = χ (b) χ (h)$ , so $t_{h}^{*} χ = χ (h) χ$ . Taking determinants gives $Norm (χ (h) χ) = Norm (χ)$ , where $Norm (ψ) = \det (m_{ψ})$ ; conjugation preserves determinants, multiplication by the scalar $χ (h)$ multiplies determinants by $χ (h)^{d}$ , and $Norm (χ)$ is invertible by [F7] applied to the invertible endomorphism $m_{χ}$ . Cancelling the unit $Norm (χ)$ yields $χ (h)^{d} = 1$ in $R$ .

5.1F2step 4.1algebra

Apply step 4.1 to $H^{D}$ , of rank $d$ . Fix a $k$ -algebra $R$ and $y \in H^{D} (R)$ , represented by a group-like element $a \in B \otimes_{k} R$ as in step 2.1. Pass to $R^{'} = B \otimes_{k} R$ and take the universal point $h \in H (R^{'})$ , which via $H ≅ H^{D D}$ is a character of $(H^{D})_{R^{'}}$ . Evaluation of that character on $y_{R^{'}}$ is exactly $a \in R^{'}$ . The determinant identity of step 4.1 consequently gives $a^{d} = 1$ . Thus the character corresponding to $[d] y$ is trivial, and $[d] y = 0$ by the representing identification of step 2.1. This proves $[d]_{H^{D}} = 0$ on every test. Duality is faithful by step 3.1, and dualizing multiplication by $d$ gives multiplication by $d$ (composition of a character with $[d]$ is its $d$ -th power), so $[d]_{H} = 0$ . The use of a universal character after base extension, rather than only characters over the original $R$ , retains infinitesimal points.

6.1F3F4F5step 3.1step 5.1algebra

Let $f : H \to H^{'}$ be a morphism of finite commutative $k$ -group schemes. Its scheme-theoretic kernel is a closed subgroup scheme, and its scheme-theoretic image is a closed subgroup scheme by [F4]; quotients of finite commutative group schemes by closed subgroup schemes are finite by [F3], [F5], since the faithfully flat quotient of the zero-dimensional scheme $H$ is a separated finite-type $k$ -scheme of dimension zero. The kernel and image identifications make the category of finite commutative $k$ -group schemes abelian: finite products and products of morphisms exist, every morphism has a kernel and a cokernel, and the fppf kernel-image identities hold. More explicitly, the induced map from $H / \ker f$ to the scheme-theoretic image of $f$ has trivial kernel, hence is a closed immersion by [F4]; it is schematically dominant by the definition of the image, hence is an isomorphism. Thus coimage equals image. Cartier duality is an additive contravariant equivalence by steps 2.1 and 3.1 (it exchanges products with coproducts because $O (H \times_{k} H^{'}) = B \otimes_{k} B^{'}$ dualizes to $B^{*} \otimes_{k} (B^{'})^{*}$ ), and an additive equivalence of abelian categories preserves kernels and cokernels, hence carries exact sequences to exact sequences with the arrows reversed.

7.1F2F6step 6.1algebra∎

For $H = μ_{n}$ one has $B = k [t] / (t^{n} - 1)$ with $Δ (t) = t \otimes t$ , so the elements $t^{j}$ , $j = 0, \dots, n - 1$ , are group-like and form a $k$ -basis; by steps 1.1 and 2.1 the dual basis elements $e_{m}$ of $B^{*}$ satisfy $e_{i} e_{j} = δ_{i j} e_{i}$ and $Δ (e_{m}) = \sum_{i + j \equiv m} e_{i} \otimes e_{j}$ . Thus $B^{*}$ is the algebra of $k$ -valued functions on $Z / n Z$ with pointwise multiplication and the comultiplication dual to addition modulo $n$ , that is, $μ_{n}^{D} ≅ (Z / n)_{k}$ .

The construction nowhere uses reducedness of $H$ : the determinant argument is over the finite free $R$ -module $B_{R}$ , and it applies also when $char k$ divides the rank $d$ .

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Abelian schemes over a base

Definition

Let $S$ be a scheme. An abelian scheme over $S$ of relative dimension $g$ is an $S$ -group scheme $f : A \to S$ in the sense of Group schemes over a base scheme such that:

$f$ is smooth (Smooth morphism of schemes);
$f$ is proper (Proper morphisms);
$f$ is locally of finite presentation (Locally finite presentation morphisms);
every geometric fibre $A_{\overset{ˉ}{s}} = A \times_{S} Spec \overset{ˉ}{κ} (s)$ is connected of dimension $g$ (Geometric properties of fibres, Scheme-theoretic fibre); equivalently, the relative dimension is constant equal to $g$ (Relative dimension of a smooth morphism at a point) and every geometric fibre is nonempty and connected.

Equivalently, an abelian scheme over $S$ is a smooth proper $S$ -group scheme whose geometric fibres are abelian varieties of dimension $g$ in the sense of Abelian varieties over a field; smoothness and properness make the fibres smooth proper connected group schemes, and conversely a family of abelian varieties of constant dimension which is a smooth proper $S$ -group scheme is an abelian scheme. The condition on geometric fibres makes $g$ locally constant on $S$ and equal to $g$ on each connected component of $S$ . Since $f$ is proper, it is separated, and the unit section $e : S \to A$ is a closed immersion, being a section of a separated morphism. The group law, inverse and unit are those of the $S$ -group scheme structure; they are automatically $S$ -morphisms of finite presentation.

No projectivity of $A$ over $S$ is asserted: an abelian scheme over a general base need not be projective over that base, and none of the results on this page assumes it. The base $S$ is not required to be Noetherian.

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An S-rational map defined after a faithfully flat smooth base change is defined

Statement

Assume AC. Let $S$ be locally Noetherian (Locally Noetherian and Noetherian schemes), let $Y$ be separated over $S$ (Separated morphism of schemes), and let $u : X ⇢ Y$ be an $S$ -rational map between smooth finite-type $S$ -schemes (S-dense open subschemes and S-rational maps). Let $f : X^{'} \to X$ be a faithfully flat morphism of smooth finite-type $S$ -schemes (Faithfully flat scheme morphism) such that the base-changed $S$ -rational map $u \circ f : X^{'} ⇢ Y$ is represented by an $S$ -morphism $X^{'} \to Y$ defined on all of $X^{'}$ . Then $u$ is represented by an $S$ -morphism $X \to Y$ defined on all of $X$ . This is BLR 2.5/5; no arbitrary base change is used.

Facts & Assumptions

Given: AC, a locally Noetherian base $S$ , a separated $S$ -scheme $Y$ , smooth finite-type $S$ -schemes $X, X^{'}$ , an $S$ -rational map $u : X ⇢ Y$ and a faithfully flat $S$ -morphism $f : X^{'} \to X$ such that $u \circ f$ is defined everywhere and equal to a morphism $g : X^{'} \to Y$ .

[F1]

An $S$ -rational map is an equivalence class of $S$ -morphisms on $S$ -dense opens, with domain of definition $dom (u)$ ; base change preserves these notions, and for separated smooth finite-type targets the domain commutes with flat base change (S-dense open subschemes and S-rational maps).

[F2]

A faithfully flat, quasi-compact, locally finitely presented morphism is a cover for fppf descent of morphisms: a morphism whose two pullbacks to $X^{'} \times_{X} X^{'}$ agree descends uniquely (Scheme morphisms satisfy fppf descent, assuming AC). Faithfully flat morphisms are surjective (Faithfully flat scheme morphism).

[F3]

For a separated target, morphisms from any source that agree on a schematically dense open are equal (Agreement on a schematically dense open, assuming AC). A smooth morphism is flat with geometrically reduced fibres (Smooth morphism of schemes). On affine charts $Spec B \to Spec A$ of a smooth finite-type map with $A$ Noetherian, a finite prime filtration of the $A$ -module $A$ tensors exactly with the flat $A$ -algebra $B$ and filters $B$ by the rings $B / p_{i} B$ . Each is flat over the domain $A / p_{i}$ , hence injects into its reduced generic fibre and is reduced. In a reduced Noetherian ring every associated prime is minimal: the ring injects into the finite product of its minimal-prime domain quotients, so the annihilator of a nonzero element is the intersection of those minimal primes where its image is nonzero; if this annihilator is prime, it equals one of those minimal primes. Flatness over $A / p_{i}$ makes every nonzero base element a nonzerodivisor, so these minimal primes contract to $p_{i}$ . The associated-prime theorem for a finite filtration then shows every associated prime of $B$ is the generic point of a component of some fibre. If an open $U$ meets every fibre densely but $\ker (B \to Γ (U, O)) \neq 0$ , this finite ideal has an associated prime; it is also associated in $B$ , so its point lies in $U$ , where the restriction kernel has zero stalk, a contradiction. Hence $U$ is schematically dense. These uses are supplied by Finite modules over Noetherian rings admit prime filtrations, A nonzero module over a Noetherian ring has an associated prime, and Associated primes in a short exact sequence.

[F4]

If $U \subseteq X$ is schematically dense and $q : T \to X$ is flat, then $q^{- 1} (U)$ is schematically dense in $T$ . Locally take $V = Spec A \subseteq X$ and an affine $W = Spec B \subseteq q^{- 1} (V)$ ; flatness makes $B$ flat over $A$ . The open $U \cap V$ is quasi-compact since $X$ is locally Noetherian. A finite principal-open cover computes $Γ (U \cap V, O)$ as a finite equalizer of localizations of $A$ ; tensoring this equalizer with flat $B$ computes $Γ (W \times_{X} U, O)$ and preserves the injection $A ↪ Γ (U \cap V, O)$ . Thus $B ↪ Γ (W \times_{X} U, O)$ , which is the required schematic density. Flatness is stable under base change (Flatness is stable under arbitrary base change).

[F5]

The given hypotheses make $f$ faithfully flat, quasi-compact, and locally of finite presentation. To see the latter two properties, work locally on affine Noetherian opens $S_{0} = Spec A \subseteq S$ . For affine charts $W = Spec C \subseteq X^{'}$ and $V = Spec B \subseteq X$ with $f (W) \subseteq V \subseteq X_{S_{0}}$ , both $B$ and $C$ are finite-type $A$ -algebras; generators of $C$ over $A$ also generate it over $B$ , so $C$ is finite type over $B$ . The ring $B$ is Noetherian because it is of finite type over the Noetherian ring $A$ , and a finite-type algebra over $B$ is finitely presented, proving local finite presentation (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras, Locally finite presentation morphisms, Every algebra of finite type over a Noetherian ring is a Noetherian ring, If $R$ is Noetherian then $R [x_{1}, \dots, x_{n}]$ is Noetherian for every $n \in N$ , Finite type is affine-local on source and target). For quasi-compactness, cover any quasi-compact open $V_{0} \subseteq X$ by finitely many affine opens $V_{i}$ each lying over an affine Noetherian open $S_{i} \subseteq S$ . Each $f^{- 1} (V_{i})$ is open in the Noetherian finite-type $S_{i}$ -scheme $X_{S_{i}}^{'}$ , hence is quasi-compact; therefore $f^{- 1} (V_{0})$ is quasi-compact. Faithful flatness is given (Faithfully flat scheme morphism).

Proof

technique · direct, descending the representative along the faithfully flat cover

1.1F3F4F1given

Let $U = dom (u)$ . Since $X$ is smooth over $S$ and $U$ is $S$ -dense, [F3] makes $U$ schematically dense in $X$ . Its pullback $U^{'} = f^{- 1} (U)$ is schematically dense in $X^{'}$ by [F4]. The everywhere-defined representative $g : X^{'} \to Y$ and the morphism $u ∣_{U} \circ f ∣_{U^{'}} : U^{'} \to Y$ agree on an $S$ -dense open by the definition of $u \circ f$ ; that open is schematically dense in the smooth scheme $U^{'}$ by [F3]. Separatedness of $Y$ and [F3] therefore give $g ∣_{U^{'}} = u ∣_{U} \circ f ∣_{U^{'}}$ .

2.1F4F3step 1.1givenalgebra

Put $X^{''} = X^{'} \times_{X} X^{'}$ and let $q : X^{''} \to X$ be the composite of either projection with $f$ . This map is flat: each projection is a base change of the flat map $f$ , hence flat by [F4], and compositions of flat morphisms are flat. The open $W = q^{- 1} (U) = f^{- 1} (U) \times_{U} f^{- 1} (U)$ is schematically dense in $X^{''}$ by [F4]. By step 1.1, the two pullbacks of $g$ agree on $W$ , where both are the composite of the common map to $U$ with $u ∣_{U}$ . As $Y$ is separated over $S$ , [F3] gives equality of these pullbacks on all of $X^{''}$ .

3.1F5F2step 2.1construct

By [F5], $f$ is faithfully flat, quasi-compact and locally finitely presented; by step 2.1 the two pullbacks of $g$ to $X^{''}$ agree. Fppf descent of morphisms [F2] therefore gives a unique $S$ -morphism $h : X \to Y$ with $h \circ f = g$ .

4.1F2step 1.1step 3.1∎

The morphism $h$ extends $u$ : on $U$ , the morphisms $h ∣_{U}$ and $u ∣_{U}$ pull back along the faithfully flat morphism $f^{- 1} (U) \to U$ to the same map $g ∣_{f^{- 1} (U)}$ by step 1.1. Uniqueness in [F2] makes them equal. Hence $h$ is a morphism on all of $X$ whose restriction to the $S$ -dense open $U$ represents $u$ , so it represents the $S$ -rational map $u$ . [F2, step 1.1, step 3.1].

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Indeterminacy of a rational map into an affine scheme is of pure codimension one

Statement

Assume AC. Let $R$ be a ring, let $Z$ be a normal Noetherian $R$ -scheme (normal noetherian ring) and let $B$ be a finitely generated $R$ -algebra (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras). Here an $R$ -rational map means an equivalence class of $R$ -morphisms on dense open subschemes, agreeing on a dense open of their intersection; on the disjoint integral components of the normal $Z$ this extends the field-case convention of Rational maps of integral finite-type schemes. For such a map $u : Z ⇢ Spec B$ the indeterminacy locus of $u$ is empty or of pure codimension one in $Z$ . In particular, if $u$ is defined at every point of height at most one, then $u$ extends uniquely to an $R$ -morphism $Z \to Spec B$ .

Facts & Assumptions

Given: AC, a ring $R$ , a normal Noetherian $R$ -scheme $Z$ , a finitely generated $R$ -algebra $B$ , and an $R$ -rational map $u : Z ⇢ Spec B$ .

[F1]

Choose $R$ -algebra generators $b_{1}, \dots, b_{n}$ of $B$ , so that $Spec B$ is a closed subscheme of $A_{R}^{n}$ cut out by the ideal of all defining relations; a morphism $Z \to Spec B$ is the same as an $R$ -algebra map $B \to Γ (Z, O_{Z})$ , equivalently a choice of $n$ regular functions satisfying those relations (Subalgebra generated by a subset, algebras of finite type, and module-finite algebras).

[F2]

A normal Noetherian domain $A$ with fraction field $K$ satisfies $A = ⋂_{ht p = 1} A_{p}$ in $K$ ; equivalently, an element of $K$ regular at every height-one point of $Spec A$ is regular (A normal Noetherian domain is the intersection of its height-one localizations, assuming AC). A rational map on an integral scheme is given by a morphism on a dense open, and two morphisms agreeing on a dense open of an integral scheme coincide (Rational maps of integral finite-type schemes).

Proof

technique · direct. The domain of the map is the common regular locus of finitely many rational functions, and each pole locus is pure codimension one by the intersection formula

1.1F1F2givenalgebra

Let $Y = Spec B$ and choose generators $b_{1}, \dots, b_{n}$ of $B$ over $R$ as in [F1]. On the dense open where $u$ is represented, the pullbacks $u^{*} (b_{i})$ are rational functions $f_{i}$ on $Z$ . On an integral affine chart $U = Spec A \subseteq Z$ with fraction field $K$ , the $f_{i}$ lie in $K$ , and a morphism $U \to Y$ is given exactly by an $R$ -algebra map $B \to A$ , i.e. by elements $f_{1}, \dots, f_{n} \in A$ satisfying every defining relation of $B$ . Since those relations vanish on the dense open where $u$ is defined, they vanish as rational functions; hence $u$ is defined at a point $z \in U$ if and only if $f_{1}, \dots, f_{n}$ all lie in the local ring $O_{Z, z}$ .

2.1F2step 1.1algebra

On an integral affine chart $U = Spec A$ , the nonregular locus of $f \in Frac A$ is $V (J_{f})$ , where $J_{f} = {a \in A : a f \in A}$ : membership in $A_{q}$ is equivalent to $J_{f}$ containing an element outside $q$ . If $q$ is a prime minimal over $J_{f}$ , then $f \notin A_{q}$ . Apply [F2] to the normal Noetherian local domain $A_{q}$ : there is a height-one prime $p A_{q}$ at which $f$ is not regular, with $p \subseteq q$ . All chains below $p$ survive localization, so $p$ has height one in $A$ . Nonregularity implies $J_{f} \subseteq p$ , and minimality of $q$ therefore gives $p = q$ . Every irreducible component of $V (J_{f})$ thus has codimension one.

3.1F2step 2.1algebra

By step 1.1 the indeterminacy locus of $u$ on $U$ is the union of the pole loci of $f_{1}, \dots, f_{n}$ . If all $f_{i}$ are regular on $U$ , this locus is empty and $u$ is a morphism on $U$ . Otherwise it is the union of finitely many closed subsets each of which is of pure codimension one by step 2.1; a finite union of pure-codimension-one closed subsets of a Noetherian scheme has all its irreducible components of codimension one, so the indeterminacy locus is of pure codimension one.

4.1F1F2step 3.1algebra∎

If $u$ is defined at every point of height at most one, then by step 2.1 no pole locus meets the height-one points of $U$ , so each pole locus is empty; thus all $f_{i}$ are regular on every affine chart, and the local morphisms $U \to Y$ glue to an $R$ -morphism $Z \to Y$ extending $u$ , unique because $Y$ is separated over $R$ and two extensions agree on the dense domain of $u$ by [F2]. Finite generation of $B$ over $R$ is used to have finitely many $b_{i}$ ; the relation ideal need not be finitely generated, so that a common regular locus can be exhibited.

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Uniqueness, weak Neron property, etale base change and local nature of Neron models

Statement

Assume AC. Let $S$ be a Dedekind scheme with function field $K$ (Locally Noetherian and Noetherian schemes) and let $X$ be a Neron model of the smooth separated finite-type $K$ -scheme $X_{K}$ (Neron models, the Neron mapping property and weak Neron models). Then:

(a) $X$ is unique up to a unique $S$ -isomorphism inducing the identity on the generic fibre;

(b) $X$ is a weak Neron model of $X_{K}$ : it satisfies the extension property for etale points at every closed point of $S$ ;

(c) for every etale morphism $S^{'} \to S$ (Étale morphism of schemes) with function field $K^{'}$ , the base change $X_{S^{'}}$ is a Neron model of $X_{K^{'}}$ ;

(d) $X$ is a Neron model over $S$ if and only if $X \times_{S} Spec O_{S, s}$ is a Neron model over $Spec O_{S, s}$ for every closed point $s \in S$ ; thus the notion is local on the base;

(e) if $X_{K}$ is a $K$ -group scheme, then its multiplication, inverse and unit extend uniquely to $X$ , making $X$ an $S$ -group scheme.

The arguments apply the mapping property directly. No converse from the weak Neron property to the full mapping property is asserted for an arbitrary scheme or for a model whose generic fibre alone carries a group law.

Facts & Assumptions

Given: AC, a Dedekind scheme $S$ with function field $K$ , a smooth separated finite-type $K$ -scheme $X_{K}$ , and a Neron model $X$ of $X_{K}$ with its Neron mapping property.

[F1]

The Neron mapping property: for every smooth $S$ -scheme $Y$ and every $K$ -morphism $Y_{K} \to X_{K}$ there is a unique $S$ -morphism $Y \to X$ extending it; the weak Neron model is defined by the extension property for etale points (Neron models, the Neron mapping property and weak Neron models).

[F2]

Etale morphisms are smooth; smooth and flat morphisms are stable under base change and composition, and etale morphisms are stable under base change (Étale morphism of schemes, Smooth morphism of schemes, Flatness is stable under arbitrary base change, Smoothness survives base change and composition).

[F3]

Two morphisms from a reduced scheme to a separated scheme that agree on a schematically dense open subscheme are equal (Agreement on a schematically dense open, assuming AC). For the generic fibre, which need not be open, the agreement assertion holds for a flat source over the integral base $S$ : the equalizer is closed by separatedness; on affine base and source charts its ideal becomes zero after tensoring with $K$ . Every element of that ideal is therefore killed by a nonzero base element, while flatness makes the source coordinate ring torsion-free, so the ideal is zero. In particular this applies to every smooth source and its overlaps (Separated morphism of schemes, Scheme-theoretic fibre, Smooth morphism of schemes).

[F4]

Morphisms between finitely presented schemes descend along filtered limits of affine base schemes, and equality descends after a later stage. In particular, $Spec O_{S, s}$ is the filtered limit of open neighbourhoods of $s$ (Finite-stage descent of finitely presented schemes and their morphisms). Smooth morphisms are locally standard smooth, so their finite-presentation presentations and invertible Jacobian minors descend to smooth neighbourhoods after shrinking (Locally standard smooth iff flat with geometrically regular fibres).

Proof

technique · direct; each assertion is an application of the mapping property to a suitable smooth source

1.1F1givenalgebra

Let $X^{'}$ be another Neron model of $X_{K}$ . Applying the mapping property of $X$ to the identity $K$ -morphism $X_{K}^{'} \to X_{K}$ gives $u : X^{'} \to X$ over $S$ , and applying the mapping property of $X^{'}$ to the identity $X_{K} \to X_{K}^{'}$ gives $v : X \to X^{'}$ ; the composites $u v$ and $v u$ extend the identity $K$ -morphisms and both sides are $S$ -morphisms, so by the uniqueness clause they are identities. This proves (a).

1.2F1F2F4givenconstruct

Let $s \in S$ be closed and $R^{'}$ an etale local $O_{S, s}$ -algebra with fraction field $K^{'}$ . It is a filtered limit of pointed etale neighbourhoods $V_{i} \to U_{i}$ with $U_{i} \subseteq S$ open and containing $s$ . The given $K^{'}$ -point of $X_{K}$ descends to the generic fibre of one such neighbourhood after passing to a later stage, by [F4]. Each $V_{i} \to U_{i} \to S$ is smooth, so the Neron property extends that stage point to $V_{i} \to X$ ; base change along the limit gives the required $R^{'}$ -point of $X$ . Thus the extension property for etale points holds at every closed point and $X$ is a weak Neron model, proving (b).

1.3F1F2algebra

For (c), let $S^{'} \to S$ be etale with function field $K^{'}$ and let $Y^{'}$ be a smooth $S^{'}$ -scheme with a $K^{'}$ -morphism $u_{K^{'}} : Y_{K^{'}}^{'} \to X_{K^{'}}$ . The composite $Y^{'} \to S^{'} \to S$ is smooth by [F2], and $u_{K^{'}}$ composed with the projection $X_{K^{'}} = X_{K} \times_{K} K^{'} \to X_{K}$ is a $K$ -morphism $Y_{K}^{'} \to X_{K}$ ; by the mapping property it extends uniquely to an $S$ -morphism $Y^{'} \to X$ . Combining this with the structure morphism $Y^{'} \to S^{'}$ gives an $S^{'}$ -morphism $Y^{'} \to X \times_{S} S^{'} = X_{S^{'}}$ extending $u_{K^{'}}$ , and uniqueness follows from the uniqueness in the $S$ -mapping property. Since $X_{S^{'}}$ is smooth separated of finite type over $S^{'}$ by [F2] and has generic fibre $X_{K^{'}}$ , it is a Neron model of $X_{K^{'}}$ .

1.4F1F4algebraconstruct

For (d), first suppose $X$ is a Neron model over $S$ . Cover a smooth $Spec O_{S, s}$ -scheme by affine standard-smooth charts. Their finite-presentation presentations and invertible Jacobian minors descend along the filtered limit of open neighbourhoods of $s$ to smooth charts over some neighbourhood, by [F4]; the generic-fibre morphism to $X_{K}$ descends at a later stage by the same finite-presentation lemma. Apply the Neron property over $S$ to each descended smooth neighbourhood and pass to the limit. These local extensions agree on overlaps by uniqueness, so they glue to an extension over $Spec O_{S, s}$ . Uniqueness follows from separatedness and density of the generic fibre.

1.5F1F3F4algebraconstruct

Conversely, suppose each localization $X_{O_{S, s}}$ is a Neron model. First take a smooth finite-type $S$ -scheme $Y$ and a $K$ -morphism $u_{K} : Y_{K} \to X_{K}$ . For each closed $s$ , the local property gives $Y_{O_{S, s}} \to X_{O_{S, s}}$ . Since $Y$ and $X$ are of finite presentation over the noetherian base, [F4] descends this map to $Y_{U} \to X_{U}$ for some open neighbourhood $U$ of $s$ , with generic restriction $u_{K}$ after a further shrinking if needed. These neighbourhoods cover the closed points of $S$ , and together with the generic point cover $S$ . The descended maps agree on overlaps because they agree on $Y_{K}$ , which is schematically dense in the flat scheme $Y$ , and $X$ is separated. They therefore glue to an extension $Y \to X$ . For an arbitrary smooth $Y / S$ , cover it by finite-type open subschemes, apply this argument on each, and glue by uniqueness. This proves the converse and the locality assertion.

2.1F1F3step 1.4algebra∎

For (e), assume $X_{K}$ is a $K$ -group scheme with multiplication $m_{K}$ , inverse $i_{K}$ and unit $e_{K}$ . Apply the mapping property to the smooth $S$ -schemes $X \times_{S} X$ , $X$ and $S$ and the $K$ -morphisms $m_{K}$ , $i_{K}$ and $e_{K}$ : this yields $S$ -morphisms $m : X \times_{S} X \to X$ , $i : X \to X$ and $e : S \to X$ extending them uniquely. Each group identity, for example associativity, is an identity between $S$ -morphisms from the reduced smooth $S$ -scheme $X \times_{S} X \times_{S} X$ to the separated $S$ -scheme $X$ ; it holds after restriction to the generic fibre, which is schematically dense by [F3], so it holds everywhere. Thus $X$ is an $S$ -group scheme.

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Two-torsion and uniqueness of the group law on a Weierstrass cubic

Statement

Assume AC. Let $k$ be a field of characteristic not $2$ or $3$ , let $a, b \in k$ with $4 a^{3} + 27 b^{2} \neq 0$ , and let $C$ be the smooth cubic $Y^{2} Z = X^{3} + a X Z^{2} + b Z^{3}$ with $O = [0 : 1 : 0]$ , regarded as an abelian variety by The chord-tangent group law on a smooth short Weierstrass cubic. Then:

(a) scheme-theoretically, $C [2]$ is the disjoint union of $O ≅ Spec k$ and $Spec k [x] / (x^{3} + a x + b)$ in the affine chart $y = 0$ ; its geometric points are $O$ and the three points $[x : 0 : 1]$ with $x^{3} + a x + b = 0$ ; in particular $C [2] (k) = {O}$ if and only if $x^{3} + a x + b$ has no root in $k$ ;

(b) if $C$ carries any abelian-variety structure with unit section $O$ , then that group law equals the chord-tangent law.

Facts & Assumptions

Given: AC, a field $k$ of characteristic not $2$ or $3$ , $a, b \in k$ with $4 a^{3} + 27 b^{2} \neq 0$ , the cubic $C : Y^{2} Z = X^{3} + a X Z^{2} + b Z^{3}$ with origin $O = [0 : 1 : 0]$ , and an abelian-variety group law $*$ on $C$ with unit $O$ .

[F1]

The chord-tangent law makes $C$ an abelian variety over $k$ with inversion $[X : Y : Z] \mapsto [X : - Y : Z]$ and with the affine formulas of The chord-tangent group law on a smooth short Weierstrass cubic; in particular the difference of the two laws is measured by the identity morphism of the underlying curve.

[F2]

A $k$ -morphism from a smooth geometrically integral group variety to an abelian variety which sends the unit to the unit is a group homomorphism (Pointed morphisms from smooth geometrically integral groups to abelian varieties are homomorphisms, assuming AC).

[F3]

Every abelian variety is commutative (A proper geometrically connected group variety is commutative), and the definitions and conventions are those of Abelian varieties over a field.

Proof

technique · direct: compute the fixed points of inversion for (a), and compare two laws by the pointed-morphism theorem for (b)

1.1F1givenalgebra

In the chord-tangent law, $2 P = O$ if and only if $P = - P$ , i.e. if and only if $P$ is fixed by the inversion $[X : Y : Z] \mapsto [X : - Y : Z]$ . For $P \neq O$ write $P = [x : y : 1]$ ; the fixed-point condition is $y = - y$ , hence $y = 0$ because $char k \neq 2$ , and such points satisfy $x^{3} + a x + b = 0$ . These equalities also compute the scheme-theoretic kernel: $[2] = 0$ is equivalent on all tests to equality of identity and inversion. On $Z = 1$ its ideal is $(2 y) = (y)$ . Near $O$ use the chart $Y = 1$ with coordinates $u = X / Y$ , $v = Z / Y$ ; inversion sends $(u, v)$ to $(- u, - v)$ , so its equalizer has ideal $(u, v)$ , defining the single reduced point $O$ .

2.1F1step 1.1algebra

The polynomial $x^{3} + a x + b$ has three distinct roots in an algebraic closure: a common root of $x^{3} + a x + b$ and its derivative $3 x^{2} + a$ would force $y = 0$ in the smoothness computation, i.e. would give $3 x^{2} = - a$ , $x^{3} = b / 2$ and hence $4 a^{3} + 27 b^{2} = 0$ , contrary to the hypothesis; so the discriminant $- (4 a^{3} + 27 b^{2})$ is nonzero. Hence $C [2] (\overset{ˉ}{k})$ consists exactly of $O$ and the three points $[x : 0 : 1]$ over the roots, and $C [2] (k) = {O}$ exactly when the cubic has no $k$ -root. These are two-torsion points, not inflection points; inflection points satisfy $3 P = O$ instead.

3.1F2F3step 2.1algebra∎

For (b), let $*$ be the given abelian-variety law with unit $O$ . The identity morphism $id : (C, \cdot) \to (C, *)$ carries the unit of the chord-tangent law to the unit of $*$ , and the source is a smooth geometrically integral group variety and the target an abelian variety, so by [F2] it is a group homomorphism; being an isomorphism of schemes, it is an isomorphism of group varieties, so the two laws coincide. The reverse implication is the same statement read backwards, and both laws are commutative by [F3].

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Prime-to-residue-characteristic Tate modules and inertia

Definition

Assume AC and DC, inherited from the multiplication and strict-henselization suppliers. Let $A$ be an abelian variety over a field $K$ (Abelian varieties over a field), fix a separable closure $K^{s e p}$ of $K$ , and let $ℓ$ be a prime different from $char K$ . For each $ν \geq 1$ the multiplication-by- $ℓ^{ν}$ endomorphism of $A$ is finite and faithfully flat and $A [ℓ^{ν}] = A \times_{[ℓ^{ν}], A, e} Spec K$ is finite locally free by Nonzero multiplication on an abelian variety is finite and faithfully flat. It is etale because the differential of multiplication by $ℓ^{ν}$ at the identity is the unit $ℓ^{ν}$ times the identity; translations give an invertible differential everywhere, and Relative Jacobian criterion with its presentation hypothesis makes multiplication etale, as is its pullback along the unit section. The $ℓ$ -adic Tate module is $T_{ℓ} (A) = {\underset{\leftarrow}{l i m}}_{ν \geq 1} A [ℓ^{ν}] (K^{s e p}),$ the inverse limit along the transition maps given by multiplication by $ℓ$ , with the underlying inverse system of finite discrete groups (Inverse systems and inverse limits of modules, The inverse limit of finite groups carries the subspace topology from the product of discrete factors). Thus an element is a sequence $(a_{ν})_{ν \geq 1}$ with $a_{ν} \in A [ℓ^{ν}] (K^{s e p})$ and $ℓ a_{ν + 1} = a_{ν}$ ; coordinatewise scalar multiplication by $Z_{ℓ} = {\underset{\leftarrow}{l i m}}_{ν} Z / ℓ^{ν} Z$ (A profinite group is a topological group isomorphic to an inverse limit of finite discrete groups) makes $T_{ℓ} (A)$ a $Z_{ℓ}$ -module. It is given the subspace topology from the product of the finite discrete torsion groups, so it is a profinite group and scalar multiplication is continuous. The absolute Galois group $Gal (K^{s e p} / K)$ acts coordinatewise on $T_{ℓ} (A)$ , and this action is $Z_{ℓ}$ -linear.

Let now $R$ be a discrete valuation ring with fraction field $K$ , and choose a strict henselization $R^{s h}$ with fraction field $K^{s h}$ embedded in $K^{s e p}$ through the fixed separable closure (Strict henselization of a DVR and smooth sections). The inertia group is $I = Gal (K^{s e p} / K^{s h}) \subseteq Gal (K^{s e p} / K) .$ The Tate module $T_{ℓ} (A)$ is unramified at $R$ when $I$ acts trivially on it. This definition specifies the action and the test; no freeness, torsion or reduction criterion is assumed.

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Strict henselian etale sections

Statement

Assume AC and DC as inherited from the stated suppliers. Let $R$ be a strictly henselian local ring with separably closed residue field $k$ (Henselian pairs and Henselian local rings) and let $H$ be a separated etale finite-type $R$ -scheme. Then:

(a) reduction induces a bijection $H (R) \to H (k)$ ;

(b) the union $H^{f i n}$ of the images of all $R$ -sections of $H$ is a finite etale $R$ -scheme isomorphic to a disjoint union of $d$ copies of $Spec R$ , where $d = ∣ H (k) ∣$ ; it contains the whole closed fibre $H_{k}$ , and its complement $H ∖ H^{f i n}$ has empty closed fibre and no $R$ -sections.

Facts & Assumptions

Given: AC and DC, a strictly henselian local ring $R$ with separably closed residue field $k$ and maximal ideal $m$ , and a separated etale finite-type $R$ -scheme $H$ .

[F1]

Etale morphisms are locally standard etale: locally on source and target, $H \to Spec R$ is $Spec (R [T] / (P))_{g}$ with $P$ monic and $P^{'}$ invertible on the localization (Étale morphisms are locally standard étale, assuming AC).

[F2]

Over a henselian local ring, a simple root of a monic polynomial lifts uniquely, and idempotents lift uniquely (A local ring is Henselian exactly when simple residue roots lift uniquely, Idempotents lift uniquely in a Henselian pair, both assuming AC); the strictly henselian property is the henselian pair condition of Henselian pairs and Henselian local rings used through these criteria.

[F3]

Here strictly henselian means henselian local with separably closed residue field. No DVR hypothesis or construction as a strict henselization is required; the henselian condition is that of Henselian pairs and Henselian local rings.

Proof

technique · direct: lift points through standard etale charts, then count the disjoint section images via the etale diagonal

1.1F1F2F3givenconstruct

Let $x \in H (k)$ and choose a standard etale chart $R [T] / (P)$ localized at $g$ around $x$ as in [F1]. The image of the chart in $Spec R$ is an open neighbourhood of the image point, which is the closed point of the local scheme $Spec R$ ; hence it is all of $Spec R$ . Write $a$ for the residue class of $T$ at $x$ ; it is a simple root of the monic polynomial $P$ because $P^{'}$ is invertible on the chart. By the simple-root lifting criterion [F2] there is a lift $a \in R$ with $P (a) = 0$ and $g (a) \notin m$ , so evaluation at $a$ defines an $R$ -section of the chart and hence of $H$ reducing to $x$ . Thus $H (R) \to H (k)$ is surjective.

2.1F1F2step 1.1algebra

Two sections $s, t$ of $H$ with equal reduction have equalizer $Eq (s, t) \subseteq Spec R$ ; the diagonal of an etale morphism is an open immersion, so it is open, and $H$ separated over $R$ makes it closed, while it contains the closed point by hypothesis; since $Spec R$ is connected (it is a local scheme), the equalizer is all of $Spec R$ , so $s = t$ . Hence reduction $H (R) \to H (k)$ is injective, and with step 1.1 it is bijective; this proves (a).

3.1F2step 2.1construct

For a section $s : Spec R \to H$ , its image is open, being the base change of the etale diagonal, and closed because $s$ is a closed immersion as a section of the separated morphism $H \to Spec R$ . Two distinct sections have disjoint images: their equalizer is open and closed by the same argument as in step 2.1, and it is empty because it is a proper closed subset of the connected scheme $Spec R$ (it misses the closed point since the sections have distinct reductions by the bijection of step 2.1). Hence the images of the $d$ sections, one for each point of the finite set $H (k)$ , form $d$ disjoint open and closed subschemes each isomorphic to $Spec R$ via $s$ .

4.1F1F3step 3.1algebra∎

Since $H$ is etale and finite type over the field $k$ , the closed fibre $H_{k}$ is a disjoint union of finitely many copies of $Spec k$ (finite separable extensions of the separably closed field $k$ are trivial), so the closed fibre is covered by the closed points $H (k)$ , and $H_{k} \subseteq H^{f i n}$ . Hence $H^{f i n}$ , the disjoint union of the $d$ section images, is finite etale over $R$ , and every section of $H$ meets $H_{k}$ and therefore lies in $H^{f i n}$ ; the complement $H ∖ H^{f i n}$ has empty closed fibre and admits no $R$ -section. This proves (b).

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The identity model of a smooth group with abelian generic fibre

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ , and let $G$ be a smooth separated finite-type $R$ -group scheme (Group schemes over a base scheme) whose generic fibre $G_{K}$ is an abelian variety (Abelian varieties over a field). Then $G^{0} = G_{K} \cup (G_{k})^{0}$ is an open $R$ -subgroup scheme of $G$ , smooth, separated and of finite type over $R$ , with geometrically connected fibres. On each geometric fibre of $G$ , the orbits of $G^{0}$ are exactly the connected components of that fibre.

Facts & Assumptions

Given: AC and DC, a DVR $R$ with fraction field $K$ and residue field $k$ , a smooth separated finite-type $R$ -group scheme $G$ with abelian generic fibre $G_{K}$ , and the identity component $(G_{k})^{0}$ of the special fibre.

[F1]

$G_{K}$ is an abelian variety, hence connected; $G_{k}$ is a smooth finite-type $k$ -group scheme with finitely many connected components, the identity component $(G_{k})^{0}$ being open and a subgroup scheme (Abelian varieties over a field, Group schemes over a base scheme).

[F2]

A connected smooth finite-type group scheme over a field is geometrically connected, and a smooth connected such group is geometrically integral; these statements propagate through field extension (Connected finite-type groups are geometrically connected, assuming AC).

[F3]

A scheme which is smooth over a discrete valuation ring is flat over it, and a closed subscheme of a scheme over a DVR whose generic and special fibres are both empty is empty; a morphism of $R$ -schemes whose restriction to the generic fibre and to the special fibre both factor through an open subscheme factors through it (Locally Noetherian and Noetherian schemes, Group schemes over a base scheme).

Proof

technique · direct: the subscheme is open by construction, closed under the group operations fibrewise, and its orbits are computed by translation

1.1F1F2givenconstruct

The set $G^{0} = G_{K} \cup (G_{k})^{0}$ is open in $G$ : by [F1], $G_{k}$ has finitely many connected components, so $G_{k} ∖ (G_{k})^{0}$ is closed in $G_{k}$ . Since the special fibre $G_{k}$ is closed in $G$ , this complement is closed in $G$ . Its complement in $G$ is exactly $G_{K} \cup (G_{k})^{0}$ , which is therefore open. It is an open subscheme, hence smooth, separated and of finite type over $R$ . Its generic fibre is the connected abelian variety $G_{K}$ and its special fibre is $(G_{k})^{0}$ , connected; by [F2] the special fibre is geometrically connected and the generic fibre is geometrically connected, so the fibres over the two points of $Spec R$ are geometrically connected.

2.1F1F3step 1.1algebra

The multiplication, inverse and unit of $G$ restrict to $G^{0}$ : the multiplication $m_{G}$ maps $G_{K} \times_{K} G_{K}$ into $G_{K}$ and $(G_{k})^{0} \times_{k} (G_{k})^{0}$ into $(G_{k})^{0}$ because $(G_{k})^{0}$ is a subgroup scheme; hence the preimage $m_{G}^{- 1} (G^{0}) \subseteq G \times_{R} G$ is an open subscheme containing both $G_{K} \times_{K} G_{K}$ and $(G_{k})^{0} \times_{k} (G_{k})^{0}$ . The complement of this preimage inside the open subscheme $G^{0} \times_{R} G^{0}$ is closed with empty generic and special fibres, hence empty by [F3]; therefore $m_{G}$ restricts to $G^{0} \times_{R} G^{0} \to G^{0}$ . The same argument with the inverse and the unit section (whose value at the closed point lies in $(G_{k})^{0}$ ) shows that $G^{0}$ is an $R$ -subgroup scheme of $G$ .

2.2F1F2step 1.1algebra

Let $\overset{ˉ}{s}$ be a geometric point of $Spec R$ . If $\overset{ˉ}{s}$ lies over the generic point, $G_{\overset{ˉ}{s}} = (G_{K})_{\overset{ˉ}{s}}$ is connected by [F2]; if $\overset{ˉ}{s}$ lies over the closed point, $G_{\overset{ˉ}{s}} = (G_{k})_{\overset{ˉ}{s}}$ and $(G^{0})_{\overset{ˉ}{s}} = ((G_{k})^{0})_{\overset{ˉ}{s}} = (G_{\overset{ˉ}{s}})^{0}$ because the identity component of a smooth group scheme over a field is geometrically connected by [F2]. Thus $G_{\overset{ˉ}{s}}^{0}$ is the identity component of the smooth group $G_{\overset{ˉ}{s}}$ .

3.1F1step 2.2algebra∎

On a geometric fibre $G_{\overset{ˉ}{s}}$ , translation by a point $x$ is an isomorphism $G_{\overset{ˉ}{s}} \to G_{\overset{ˉ}{s}}$ carrying the identity component onto the connected component of $x$ ; since $G_{\overset{ˉ}{s}}^{0}$ is the identity component by step 2.2, the orbit of $x$ under $G_{\overset{ˉ}{s}}^{0}$ is exactly the connected component of $x$ . Hence the orbits of $G^{0}$ on each geometric fibre are the connected components, as claimed.

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Dilatations and defect computation

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ , residue field $k$ , uniformizer $π$ , and let $R^{s h}$ be a strict henselization. Let $X$ be a finite-type flat $R$ -scheme with smooth generic fibre of relative dimension $d$ , and let $Y \subseteq X_{k}$ be a closed subscheme.

(a) The $π$ -chart of the blowup ${Bl}_{Y} (X)$ is flat over $R$ and universally receives a unique $R$ -morphism over $X$ from every given $R$ -morphism $B \to X$ with $B$ flat over $R$ whose special morphism $B_{k} \to X_{k}$ factors through $Y$ ; it is called the dilatation of $X$ along $Y$ . Dilatations commute with unramified flat base change of DVRs and with products. A closed immersion $X_{1} ↪ X$ of flat $R$ -schemes, with centre $Y_{1} = Y \times_{X} X_{1}$ , induces a closed immersion of the corresponding dilatations; this is not a claim that arbitrary closed base change gives a cartesian square.

(b) For a section $a : Spec R \to X$ , the defect $δ (a)$ is the length of the torsion submodule of $a^{*} Ω_{X / R}$ ; it vanishes if and only if $X$ is smooth along $a$ , and for smooth generic fibre it equals the minimum valuation of the maximal-rank Jacobian minors of a standard presentation of generic codimension, and is bounded uniformly over all $a \in X (R^{s h})$ .

(c) A morphism between smooth $R$ -schemes of the same relative dimension is etale exactly at the points where its relative differential determinant is invertible; in particular $δ (a) = 0$ means that the special-fibre cotangent space at the rational specialization of $a$ has dimension $d$ .

Facts & Assumptions

Given: AC and DC, a DVR $R$ with uniformizer $π$ , fraction field $K$ and residue field $k$ , a strict henselization $R^{s h}$ , a flat finite-type $R$ -scheme $X$ with smooth generic fibre of dimension $d$ , a closed subscheme $Y \subseteq X_{k}$ , and a section $a$ of $X$ .

[F1]

The standard charts of an affine blowup present the $π$ -chart as $A [I / π]$ , the quotient of $A [t_{j}] / (π t_{j} - g_{j})$ by its $π$ -power torsion; the Rees-Proj blowup is locally H-projective over $X$ , hence proper, and the valuative criterion gives unique lifting of sections (Affine blowup standard charts and overlaps, Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains, Blowups of finite type ideals are locally H-projective, and proper, Valuative criterion for properness, the last three assuming AC).

[F2]

Local fibre dimension is upper semicontinuous and bounded above by tangent dimension (Local fibre-dimension bound from polynomial quasi-finiteness); smooth morphisms have locally free differentials of rank the relative dimension, and the Jacobian criterion detects standard smooth charts by unit minors (Differentials of a smooth morphism, Relative Jacobian criterion with its presentation hypothesis, Locally standard smooth iff flat with geometrically regular fibres, Etale morphisms are the formally etale morphisms locally of finite presentation).

[F3]

Over a DVR, flat is equivalent to torsion-free; smooth total spaces are regular, regular local rings are UFDs and the regular local rings of smooth fibres have the stated divisorial properties; the normal-domain intersection formula gives Hartogs extension in codimension one (Over a principal ideal domain flatness is equivalent to torsion-freeness, Every DVR is a PID, Regularity ascends and descends along a flat local homomorphism, Regular local rings are unique factorization domains, A normal Noetherian domain is the intersection of its height-one localizations).

Proof

technique · direct: compute the dilatation chart, prove the relative etale criterion, and read off the defect

1.1F1F3givenconstruct

For an affine chart $Spec A \subseteq X$ with ideal $I \subseteq A$ cutting out $Y$ on the special fibre and containing $π$ , the $π$ -chart of the blowup has coordinate ring $A [I / π] = A [t_{j} : j] / (π t_{j} - g_{j})$ modulo its $π$ -power torsion, by [F1]. This ring is $π$ -torsion-free by construction, hence flat over the DVR $R$ by [F3]. If $B$ is flat over $R$ and the special morphism $B_{k} \to X_{k}$ factors through $Y$ , then the images of the generators $g_{j}$ in $B$ are divisible by $π$ : they vanish modulo $π$ because the factorization makes them lie in $I B$ , so $g_{j} = π h_{j}$ with $h_{j} \in B$ unique, multiplication by $π$ being injective on the flat, hence $π$ -torsion-free ring $B$ . Sending $t_{j} \mapsto h_{j}$ kills all torsion and defines the unique $R$ -morphism from $Spec B$ to the $π$ -chart; the blowup is locally H-projective over $X$ , hence proper, and the valuative criterion gives the unique lifting of sections.

2.1F1F3step 1.1algebra

For an unramified flat extension of DVRs $R \to R^{'}$ , $π$ remains a uniformizer up to a unit. The inclusion $A [I / π] \subseteq A [1 / π]$ stays injective after tensoring with $R^{'}$ , and its image is precisely the subalgebra generated by $A \otimes_{R} R^{'}$ and the images $g_{j} / π$ . Thus it is the dilatation algebra after base change. For two flat models, the tensor product of their dilatation algebras is flat over $R$ and is generated over $A_{1} \otimes_{R} A_{2}$ by the fractions from both centre ideals; the product centre has ideal $I_{1} (A_{1} \otimes_{R} A_{2}) + I_{2} (A_{1} \otimes_{R} A_{2})$ . The universal property checked componentwise therefore identifies this tensor product with the product-centre dilatation. Finally, for a flat closed subscheme with affine ring $A / J$ , the homomorphism $A [I / π] \to (A / J) [\overset{ˉ}{I} / π]$ , where $\overset{ˉ}{I}$ is the image of $I$ , is surjective: the target is generated by the images of $A$ and $g_{j} / π$ , and $π$ -power torsion maps to zero. These affine surjections glue to the asserted closed immersion. The unique local factorizations in step 1.1 likewise glue for any flat source scheme $B$ .

3.1F2step 2.1algebra

Let $f : V \to W$ be a morphism between smooth $R$ -schemes of equal relative dimension $d$ . Near $f (v)$ choose etale coordinates $W \to A_{R}^{d}$ , using a unit minor of a standard smooth presentation, and denote the pulled-back coordinate functions on $V$ by $f_{1}, \dots, f_{d}$ . If the relative differential determinant of $f$ is a unit at $v$ , the $d f_{i}$ form a basis of $Ω_{V / R}$ there. In a standard smooth presentation of $V$ in $n$ variables, append the $d$ graph equations $T_{i} - f_{i}$ to its $n - d$ relation equations. Their differentials have a unit $n \times n$ minor, so the composite $V \to A_{R}^{d}$ is etale at $v$ by [F2]. Because $W \to A_{R}^{d}$ is etale, this implies $f$ is etale: for a nilpotent lifting problem over $W$ , unique lifting over $A_{R}^{d}$ first gives the lift into $V$ , and formal unramifiedness of $W \to A_{R}^{d}$ forces its composite into $W$ to be the prescribed map. Local finite presentation then gives etaleness by [F2]. Conversely, if $f$ is etale, the same lifting property identifies its relative differential map with an isomorphism of the two locally free rank- $d$ modules, so its determinant is a unit.

4.1F2F3step 3.1algebra

For a section $a$ of $X$ with smooth generic fibre of dimension $d$ , the module $a^{*} Ω_{X / R}$ is finitely generated over the DVR, hence the direct sum of a free part and a torsion part, and $δ (a)$ is the length of that torsion part. If $δ (a) = 0$ , the differential vector space at the rational specialization of $a$ has dimension $d$ ; the generic section specializes to the special section, so the upper semicontinuity of local fibre dimension [F2] gives special local dimension at least $d$ , while tangent dimension bounds it above by $d$ . Choosing $n - d$ local equations with independent differentials exhibits a standard smooth ambient $Z$ of relative dimension $d$ containing $X$ locally along $a$ ; flatness makes the local dimension of $X$ equal to its special-fibre dimension plus one, namely $d + 1$ , which is the local dimension of the smooth ambient $Z$ at that rational point. A regular local domain has no nonzero ideal whose quotient has the same dimension, so the defining ideal of $X$ in $Z$ is zero locally. Hence $X$ agrees with the smooth ambient near the specialization and $a$ factors through the smooth locus. Hence $X$ is smooth along $a$ , and conversely smoothness makes $a^{*} Ω$ locally free, so its torsion vanishes.

5.1F2step 4.1algebra

Choose a finite affine cover $Spec A_{α}$ of $X$ and presentations $A_{α} = R [T_{1}, \dots, T_{n_{α}}] / (f_{1}, \dots, f_{m_{α}})$ . Put $q_{α} = n_{α} - d$ . A section whose specialization lies in this chart factors through the chart, since an open subset of $Spec R$ containing its closed point is the whole spectrum. Evaluation of the Jacobian gives a presentation $R^{m_{α}} \to R^{n_{α}} \to a^{*} Ω_{X / R} \to 0$ of generic rank $q_{α}$ . Smith normal form over the DVR shows that the torsion length is the sum of the valuations of its $q_{α}$ nonzero diagonal entries; equivalently it is the minimum valuation of the $q_{α} \times q_{α}$ minors. This proves the asserted numerical formula, with the size-zero minor interpreted as $1$ .

6.1F2step 3.1step 4.1step 5.1algebra∎

Let $J_{α} \subseteq A_{α}$ be the ideal generated by these minors before evaluation. Smoothness of the pure relative-dimension- $d$ generic fibre says the Jacobian has rank $q_{α}$ at every generic-fibre point, hence $J_{α} A_{α} [1 / π] = A_{α} [1 / π]$ . Expressing $1$ as a finite linear combination of minors over $A_{α} [1 / π]$ and clearing denominators gives $π^{N_{α}} \in J_{α}$ for some $N_{α} \geq 0$ . After evaluating any $R^{s h}$ -section in this chart, the minor ideal therefore contains $π^{N_{α}}$ , so step 5.1 gives $δ (a) \leq N_{α}$ . A section over $R^{s h}$ factors through a chart containing its specialization just as above, and the finite maximum $\max_{α} N_{α}$ gives the uniform bound. Finally, step 4.1 identifies defect zero with smoothness along the section and with a $d$ -dimensional special cotangent space; step 3.1 supplies the determinant criterion in (c).

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Affine codimension-one neighbourhoods and divisors

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results.

(a) On a finite-type separated normal scheme over an affine Noetherian base, finitely many points of codimension at most one lie in a single affine open subscheme.

(b) For a normal Noetherian separated scheme $X$ and a dense affine open subscheme $U \subseteq X$ , the complement $X ∖ U$ has pure codimension one in $X$ ; if $X$ is regular in addition, its reduced support is an effective Cartier divisor. If $X$ is flat over a discrete valuation ring and $U$ meets every irreducible component of the special fibre, then $X ∖ U$ is the closure of its generic fibre complement, so it contains no special-fibre component.

Facts & Assumptions

Given: AC and DC, an affine Noetherian base ring $R$ and a finite-type separated normal $R$ -scheme $X$ with points $x_{1}, \dots, x_{n}$ of codimension at most one.

[F1]

Valuative uniqueness holds for separated schemes: a valuation ring admits at most one centre on a separated scheme dominating a given centre (Valuative uniqueness detects separatedness, Valuative criterion for properness); at distinct codimension-one points this identifies the normal local rings $O_{X, x_{i}}$ with distinct DVRs in the function field.

[F2]

Hartogs for normal Noetherian domains: a rational function on a normal Noetherian domain which is regular at every height-one point is regular (A normal Noetherian domain is the intersection of its height-one localizations, assuming AC); the scheme Zariski Main Theorem and the regular-local UFD property give the corresponding divisorial statements (Scheme Zariski Main factorization for separated quasi-finite morphisms, Regular local rings are unique factorization domains, Regularity ascends and descends along a flat local homomorphism, Locally standard smooth iff flat with geometrically regular fibres).

[F3]

Rational sections of line bundles correspond to Cartier divisors, and finite prime avoidance is available (Rational sections of line bundles are Cartier divisors, An ideal contained in a finite union of prime ideals lies in one of them).

Proof

technique · direct. Approximation for finitely many inequivalent valuations produces a common affine neighbourhood of the given points, and Hartogs controls complements of dense affine opens

1.1F1givenalgebra

Reduce to a connected normal component of $X$ . Its generic point lies in every nonempty affine open, so discard it from the list; if the list becomes empty, any affine open suffices. Remove repeated points. For the remaining codimension-one points put $V_{i} = O_{X, x_{i}}$ , viewed as rank-one valuation rings in the common function field $L$ . The $V_{i}$ are pairwise distinct: if two points $x_{i} \neq x_{j}$ gave centres of the same valuation of $L$ , both would dominate the same valuation ring and separatedness would force $x_{i} = x_{j}$ by [F1]. Two distinct rank-one valuation rings in $L$ are incomparable: if $V \subseteq W$ and the uniformizer $π$ of $V$ is invertible in $W$ , then $L = V [1 / π] \subseteq W$ , so $W$ is a field, and otherwise every $x \notin V$ has inverse in $π V$ and cannot lie in $W$ ; hence inclusion forces equality.

2.1F1step 1.1construct

Fix $i$ and, for each $j \neq i$ , choose $y_{i j} \in V_{i} ∖ V_{j}$ . Multiplying a sufficiently high power of $y_{i j}$ by a uniformizer of $V_{i}$ gives $z_{i j}$ with $v_{i} (z_{i j}) > 0$ and $v_{j} (z_{i j}) < 0$ . If there is only one valuation, take $h_{i}$ to be its uniformizer; otherwise begin with one $z_{i j}$ and construct $h_{i}$ successively: to add a new index $j$ , replace the preceding sum $h$ by $h + z_{i j}^{M}$ . Choose $M$ so large that its new term has strictly smaller valuation than $h$ at $j$ and at every earlier index where $z_{i j}$ has negative valuation. At an earlier index where that valuation is nonnegative, the old negative valuation persists. Thus cancellation is excluded at every required index, and $v_{i} (h_{i}) > 0$ , $v_{j} (h_{i}) < 0$ for all $j \neq i$ . Put $e_{i} = 1 / (1 + h_{i}^{N})$ . Increasing $N$ makes $v_{i} (e_{i} - 1)$ and all $v_{j} (e_{i})$ for $j \neq i$ arbitrarily large. For prescribed targets $c_{i} \in L$ , the sum $\sum_{i} e_{i} c_{i}$ consequently approximates $c_{j}$ at every $v_{j}$ to any fixed finite precision.

3.1F3step 2.1algebra

Let $C = ⋂_{i} V_{i}$ . The residue map $C \to κ (V_{i})$ is surjective: approximate a representative in $V_{i}$ modulo its maximal ideal and approximate zero at all other valuations, using step 2.1. Thus its kernel $m_{i} = {c \in C : v_{i} (c) > 0}$ is maximal. These are all the maximal ideals: an element of $C$ is invertible exactly when all its valuations vanish, so the nonunits are the union of the $m_{i}$ , and finite prime avoidance [F3] makes every maximal ideal one of the $m_{i}$ . The approximation of step 2.1 separates the $m_{i}$ , realizes arbitrary prescribed residues, and shows $C_{m_{i}} = V_{i}$ : for $x \in V_{i}$ one approximates $1$ at $i$ and $0$ to high order at the other indices by some $t \in C$ , and then $t x \in C$ with $t \notin m_{i}$ , so $x = (t x) / t \in C_{m_{i}}$ , the reverse inclusion being immediate.

4.1step 3.1constructalgebra

Choose affine charts $W_{i} = Spec B_{i}$ around $x_{i}$ and finite $R$ -algebra generators $b_{i l}$ of $B_{i}$ . By step 3.1 write $b_{i l} = a_{i l} / s_{i l}$ with $a_{i l}, s_{i l} \in C$ and $s_{i l} \notin m_{i}$ . Let $C_{0} \subseteq C$ be the $R$ -algebra generated by all these numerators and denominators, and put $s_{i} = \prod_{l} s_{i l}$ , so $B_{i} \subseteq (C_{0})_{s_{i}}$ inside $L$ . Each of the finitely many generators of $C_{0}$ lies in $(B_{i})_{p_{i}} = V_{i}$ , where $p_{i}$ represents $x_{i}$ . Writing those generators as fractions in $(B_{i})_{p_{i}}$ and multiplying their denominators gives $t_{i} \in B_{i} ∖ p_{i}$ with $C_{0} \subseteq (B_{i})_{t_{i}}$ . Since $t_{i} \in (C_{0})_{s_{i}}$ , write $t_{i} = d_{i} / s_{i}^{N_{i}}$ with $d_{i} \in C_{0}$ ; both $s_{i}$ and $d_{i}$ are units in $V_{i}$ . The two inclusions just constructed induce mutually inverse homomorphisms, inside $L$ , between $(C_{0})_{s_{i} d_{i}}$ and $(B_{i})_{t_{i}} [1 / s_{i}]$ : $B_{i}$ lies in $(C_{0})_{s_{i}}$ and $t_{i}$ becomes invertible after inverting $d_{i}$ , while $C_{0}$ lies in $(B_{i})_{t_{i}}$ and $d_{i} = t_{i} s_{i}^{N_{i}}$ becomes invertible after inverting $s_{i}$ . All defining relations and both inverse identities hold because these are subrings of the same field. To identify an actual principal open of $W_{i}$ , write $s_{i} = r_{i} / t_{i}^{M_{i}}$ in $(B_{i})_{t_{i}}$ ; then $r_{i} \notin p_{i}$ and $(B_{i})_{t_{i}} [1 / s_{i}] = (B_{i})_{t_{i} r_{i}}$ . Thus $U_{i} = D_{W_{i}} (t_{i} r_{i})$ contains $x_{i}$ and is isomorphic to $D_{Spec C_{0}} (s_{i} d_{i})$ .

5.1F1F3step 4.1construct

The inverse morphisms $D (s_{i} d_{i}) \to U_{i} \subseteq X$ agree on every overlap: they agree at the common generic point, their source is integral, and separatedness of $X$ makes their equalizer closed. Their maps to $Spec C_{0}$ likewise agree on overlaps of the $U_{i}$ , since all are defined by the inclusion $C_{0} \subseteq L$ . Consequently these isomorphisms glue to an isomorphism from the open $W = ⋃_{i} U_{i} \subseteq X$ onto the open $⋃_{i} D (s_{i} d_{i}) \subseteq Spec C_{0}$ . Let $J$ define its closed complement. For every prime $q_{i}$ corresponding to $x_{i}$ , $J ⊈ q_{i}$ ; finite prime avoidance gives $f \in J ∖ ⋃_{i} q_{i}$ . Then $D (f) \subseteq W$ is affine and contains all $x_{i}$ . A normal Noetherian scheme has finitely many disjoint open and closed integral components; applying this construction on each component with a prescribed point and taking the finite disjoint union gives (a), including any generic points discarded in step 1.1.

6.1F2step 5.1algebra

For (b), work on one integral component of $X$ , with function field $L$ , and write $U = Spec A$ there. If an irreducible component of the closed boundary has generic point $z$ of codimension at least two, choose an affine normal chart $V = Spec B$ containing $z$ and avoiding every other boundary component. Every height-one point of $V$ then belongs to $U$ . For each $a \in A \subseteq L$ , its restriction to $U \cap V$ is regular at all these points, so [F2] places $a$ in $B$ . This gives a single ring homomorphism $A \to B$ : sums, products, the unit and every relation are preserved inside $L$ . Thus it defines an actual morphism $h : V \to U$ , with no finite-generation assumption on $A$ needed. On the dense open $U \cap V$ it is the identity inclusion into $U$ ; separatedness of $X$ makes the composite $V \overset{h}{\to} U ↪ X$ equal to the inclusion $V ↪ X$ everywhere. Its image would put $z$ in $U$ , a contradiction. Hence every boundary component has codimension one. If $X$ is regular, at each point its finitely many boundary prime ideals are principal in the regular local UFD. Their intersection is generated by the product of their distinct prime generators, a nonzerodivisor; these reduced ideals glue to the reduced boundary subscheme, making it an effective Cartier divisor. The same argument on the finitely many normal components proves (b) for general $X$ .

7.1F2step 6.1algebra∎

Finally let $X$ be flat over a discrete valuation ring and let $U$ meet every irreducible component of the special fibre. A prime divisor $P$ contained in the boundary $X ∖ U$ and dominating the base would meet the generic fibre; a prime divisor contained in the special fibre would be a component of it, which is excluded by the fibre-density hypothesis. Hence every boundary prime meets the generic fibre, so the closure of the generic complement $X_{K} ∖ U_{K}$ contains the whole boundary and is contained in it by closedness; the boundary is therefore the closure of its generic complement and contains no special-fibre component.

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Base change and products of abelian schemes

Statement

Assume AC, inherited from the smoothness and properness stability suppliers. Let $A \to S$ and $B \to S$ be abelian schemes over $S$ of relative dimensions $g_{A}, g_{B}$ (Abelian schemes over a base), and let $S^{'} \to S$ be a morphism (Base change of objects, morphisms and properties). Then:

(a) the base change $A_{S^{'}} = A \times_{S} S^{'} \to S^{'}$ is an abelian scheme of relative dimension $g_{A}$ , with $S^{'}$ -group structure induced by that of $A$ , and for $s^{'} \in S^{'}$ with image $s \in S$ the fibre $(A_{S^{'}})_{s^{'}}$ is the base change $A_{s} \times_{κ (s)} κ (s^{'})$ of abelian varieties (Fibres after base change);

(b) the product $A \times_{S} B \to S$ is an abelian scheme of relative dimension $g_{A} + g_{B}$ with the product group law;

(c) kernels of homomorphisms of abelian schemes commute with arbitrary base change by their fibre-product definition, and the base change of a finite locally free subgroup scheme is again finite locally free of the same rank.

Scheme-theoretic images are not asserted to commute with arbitrary base change.

Facts & Assumptions

Given: AC and abelian schemes $A \to S$ , $B \to S$ of relative dimensions $g_{A}, g_{B}$ , and a morphism $S^{'} \to S$ .

[F1]

An abelian scheme is a smooth proper finitely presented $S$ -group scheme with connected geometric fibres of constant dimension (Abelian schemes over a base); assuming AC for the named stability suppliers, smoothness, properness, local finite presentation, flatness and relative dimension are stable under base change and preserved by products (Flatness is stable under arbitrary base change, Smoothness survives base change and composition, Properness survives arbitrary base change, Local finiteness conditions under base change, Relative dimension of a smooth morphism at a point).

[F2]

Fibres of a base change are computed by the fibre product of fibres (Fibres after base change); the group operations of an $S$ -group scheme base change to give the induced $S^{'}$ -group structure, and products inherit the componentwise group law.

Proof

technique · direct: check the defining properties of an abelian scheme after base change and for products

1.1F1F2givenalgebra

The base change $A_{S^{'}} \to S^{'}$ is smooth, proper and locally of finite presentation by the stability statements in [F1]; its geometric fibres are base changes of geometric fibres of $A \to S$ , hence nonempty and connected of dimension $g_{A}$ , and the relative dimension is $g_{A}$ . The base-changed group operations give an $S^{'}$ -group scheme structure. For a point $s^{'} \mapsto s$ the fibre identification is the base-change compatibility of fibres in [F2].

1.2F1givenalgebra

For the product, $A \times_{S} B \to S$ is smooth, proper and locally of finite presentation, its geometric fibres are products of nonempty connected smooth proper schemes over an algebraically closed field, and a product of nonempty connected schemes over an algebraically closed field is connected (indeed the product of geometrically connected schemes over a field with a rational point in the appropriate sense is geometrically connected); the relative dimension is $g_{A} + g_{B}$ . The componentwise group law makes it an $S$ -group scheme. This proves (b).

2.1F1F2step 1.1algebra∎

Kernels of $S$ -group scheme homomorphisms are defined by the fibre product with the unit section, so they commute with arbitrary base change by associativity of fibre products, as asserted in (c); a finite locally free subgroup scheme of rank $r$ pulls back to a finite locally free subgroup scheme of the same rank because finite locally free modules and isomorphisms pull back along the base change. Scheme-theoretic images are not claimed to commute with arbitrary base change, and nothing here asserts that.

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Weil's extension theorem for rational maps into smooth separated group schemes

Statement

Assume the Axiom of Choice and the Axiom of Dependent Choice, inherited from the descent and purity suppliers. Let $S$ be a regular Noetherian base scheme (Locally Noetherian and Noetherian schemes), let $Z$ be a smooth $S$ -scheme, and let $G$ be a smooth separated $S$ -group scheme of finite type (Group schemes over a base scheme, Smooth morphism of schemes, Separated morphism of schemes). If an $S$ -rational map $u : Z ⇢ G$ (S-dense open subschemes and S-rational maps) is defined in codimension at most one, that is, at every height-one point of $Z$ , then $u$ is defined everywhere and extends uniquely to an $S$ -morphism $Z \to G$ .

Facts & Assumptions

Given: AC and DC, a regular Noetherian base $S$ , a smooth $S$ -scheme $Z$ , a smooth separated finite-type $S$ -group scheme $G$ , and an $S$ -rational map $u : Z ⇢ G$ with domain $U$ containing every height-one point of $Z$ .

[F1]

Domains of $S$ -rational maps and their behaviour under flat and faithfully flat base change are S-dense open subschemes and S-rational maps and An S-rational map defined after a faithfully flat smooth base change is defined.

[F2]

The indeterminacy locus of a rational map into an affine scheme over a normal Noetherian base is empty or of pure codimension one (Indeterminacy of a rational map into an affine scheme is of pure codimension one, assuming AC).

[F3]

A regular local ring is a UFD and a normal domain, Krull's principal ideal theorem holds, and smoothness over a regular base yields regular local rings of the total space with geometrically regular fibres (Regular local rings are unique factorization domains, regular local rings are normal, Krull's principal ideal theorem, Regularity ascends and descends along a flat local homomorphism, Locally standard smooth iff flat with geometrically regular fibres, Fibres of a smooth morphism are smooth, Fibre product of schemes).

Proof

technique · direct: analyse the difference map near the diagonal, then descend along a faithfully flat projection

1.1F1F3givenconstruct

Work locally on $S$ and $Z$ , with $S$ affine and $Z$ of finite type, so the finite-type descent lemma [F1] applies. The total spaces $Z$ and $Z \times_{S} Z$ are regular by [F3]. Form $v (z_{1}, z_{2}) = u (z_{1}) u (z_{2})^{- 1}$ on $U \times_{S} U$ and let $V$ be its maximal domain. On $V \cap Δ_{Z}$ , $v$ is the unit: it is the unit on the dense open $U \subset Δ_{Z}$ , so separatedness gives equality wherever both morphisms are defined.

2.1F1F2step 1.1algebra

Suppose $x \in Δ_{Z} ∖ V$ , with image $s \in S$ . Choose an affine open $H \subset G$ containing $e (s)$ and shrink around $s$ so $e$ lands in $H$ . Choose an integral regular affine neighbourhood $W$ of $x$ in $Z \times_{S} Z$ . The open $V \cap W \cap v^{- 1} (H)$ is nonempty: every neighbourhood of $x$ meets $Δ_{Z} \cap U$ , where $v = e$ . It is therefore dense in $W$ and represents an ordinary rational map $v^{'} : W ⇢ H$ . Let $V^{'}$ be its maximal domain. Then $V^{'} \subset V \cap W$ , and $V^{'} \cap Δ_{Z} = V \cap W \cap Δ_{Z}$ , since at a diagonal point where $v$ is defined its value lies in $H$ . By [F2], $F^{'} = W ∖ V^{'}$ is pure codimension one. Its intersection with the diagonal is contained in $Δ_{Z} ∖ U$ , which has codimension at least two in $Δ_{Z}$ .

3.1F2F3step 2.1algebra

At $x$ the reduced support of $F^{'}$ is cut out by a product $f$ of prime elements in the regular local UFD $O_{W, x}$ . Its restriction to the regular local ring of the diagonal is nonzero, because $U$ is dense in the diagonal, and is a nonunit, because $x \in F^{'}$ . The principal ideal theorem [F3] then gives a codimension-one component of $F^{'} \cap Δ_{Z}$ locally at $x$ , contradicting step 2.1. Hence $V$ contains the diagonal.

4.1F1step 3.1algebra∎

Put $Z^{'} = V \cap (Z \times_{S} U)$ . Its first projection $f : Z^{'} \to Z$ is flat, as a restriction of a smooth projection. For every geometric point $z$ of $Z$ , the open $V_{z}$ in the corresponding fibre of the second factor contains the diagonal point and is nonempty; it meets the fibrewise dense open $U$ , so $f$ is surjective. Thus $f$ is faithfully flat, and $(z_{1}, z_{2}) \mapsto v (z_{1}, z_{2}) u (z_{2})$ on $Z^{'}$ represents $u \circ f$ everywhere. Both $Z^{'}$ and $Z$ are smooth of finite type over the locally Noetherian base in this local calculation, so [F1] descends it to a morphism $Z \to G$ . These local extensions glue uniquely, since they agree on the schematically dense domain of $u$ and $G$ is separated.

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Good reduction of an abelian variety over a Dedekind scheme

Definition

Let $S$ be a Dedekind scheme (Dedekind domains) with function field $K$ , and let $A_{K}$ be an abelian variety over $K$ (Abelian varieties over a field). One says that $A_{K}$ has good reduction over $S$ if there exists an abelian scheme $A \to S$ (Abelian schemes over a base) together with an isomorphism $A \times_{S} Spec K ≅ A_{K}$ of $K$ -schemes; such an $A$ is an abelian scheme model of $A_{K}$ over $S$ . For a discrete valuation ring $R$ with fraction field $K$ (Discrete valuation rings) this is the local notion at its closed point, and $A_{K}$ has potential good reduction if there is a finite extension $K^{'} / K$ such that $A_{K} \otimes_{K} K^{'}$ has good reduction over the normalization of $R$ in $K^{'}$ .

For a closed point $s \in S$ , the reduction of $A$ at $s$ is $A_{s} = A \times_{S} Spec κ (s)$ , which is an abelian variety over the residue field $κ (s)$ of dimension $\dim A_{K}$ . Good reduction is a property of the pair $(A_{K}, S)$ ; the definition asserts no existence statement, and in particular no claim is made that every abelian variety has good reduction.

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The rigidified relative Picard functor and the dual abelian variety

Definition

Assume AC for the site-level sheafification construction. Let $f : A \to S$ be an abelian scheme (Abelian schemes over a base) with unit section $e : S \to A$ . For an $S$ -scheme $T$ write $A_{T} = A \times_{S} T$ and $e_{T} = e \times_{S} {id}_{T}$ ; a rigidified line bundle on $A_{T}$ is a pair $(L, α)$ consisting of an invertible sheaf $L$ on $A_{T}$ (Invertible sheaves) and a trivialisation $α : O_{T} \to e_{T}^{*} L$ along the unit section.

The rigidified relative Picard functor $P_{A / S, e}$ is the fppf sheaf on the category of $S$ -schemes associated (Fppf sheaves of sets and sheafification, Sheafification exists for the fppf site) with the functor $T ⟼ {isomorphism classes of rigidified line bundles on A_{T}},$ with group law given by tensor product of rigidified line bundles, identity the trivially rigidified structure sheaf and inverse by the dual pairing. An $S$ -scheme representing $P_{A / S, e}$ is called a relative Picard scheme of $A / S$ ; its identity component is written ${Pic}_{A / S}^{0}$ , and when this exists and is an abelian scheme over $S$ it is called the dual abelian variety $\hat{A}$ of $A$ . A Poincare sheaf is the universal rigidified invertible sheaf $P$ on $A \times_{S} \hat{A}$ .

The algebraically trivial subfunctor of $P_{A / S, e}$ consists of the classes whose geometric-fibre restrictions are algebraically equivalent to zero, where algebraic equivalence is the equivalence relation generated by differences of fibres in connected finite-type families of line bundles; this definition of the subfunctor does not presuppose a representing scheme, and once a Picard scheme exists the appropriate identity-component theorem identifies the subfunctor with ${Pic}_{A / S}^{0}$ . The notation "degree zero" here refers to this subfunctor and is not numerical degree on $A$ when $\dim A \geq 2$ . For $A$ over a field $k$ the dual $\hat{A}$ means a representing abelian variety for this subfunctor; representability of $P_{A / S, e}$ , the existence of ${Pic}^{0}$ and the Poincare sheaf remain claims of the commissioned theorem, and no link from this definition to that theorem is a dependency. The functor is considered with the fppf topology; all test objects are arbitrary $S$ -schemes, and flatness and local finite presentation enter only through the definitions of abelian schemes and invertible sheaves used above (Flat morphism of schemes, Locally finite presentation morphisms, Group schemes over a base scheme).

Sheafification is taken on a fixed set-sized big fppf site containing the test schemes in use, as in the cited construction; the convention imposes no finite-type or reducedness restriction on those tests. AC selects representatives and equality refinements in the plus construction, not line bundles or a representing Picard scheme.

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Universal structure-sheaf sections of an abelian scheme

Statement

Assume AC and DC as inherited from coherent cohomology and base change. Let $f : A \to S$ be an abelian scheme (Abelian schemes over a base). For every morphism $T \to S$ the unit map $O_{T} \to f_{T, *} O_{A_{T}}$ of the base-changed abelian scheme $A_{T} = A \times_{S} T \to T$ is an isomorphism, with inverse given by evaluation along the identity section; consequently every geometric fibre has $H^{0} (A_{s}, O) = k (s)$ and $f_{*} O_{A} ≅ O_{S}$ universally.

Facts & Assumptions

Given: AC and DC, an abelian scheme $f : A \to S$ , a morphism $T \to S$ and the base change $A_{T} \to T$ .

[F1]

An abelian scheme is smooth, proper and finitely presented with connected geometric fibres (Abelian schemes over a base).

[F2]

A proper geometrically integral scheme over a field has global functions equal to the field (Global functions on proper integral schemes form a finite extension of the base field).

[F3]

For a proper flat finitely presented morphism and a finitely presented flat sheaf, the higher direct images form a perfect complex compatible with base change; the finite-free base-change criterion turns surjectivity of the degree-zero fibre map into universal base change and finite local freeness (Universal finite projective cohomology complex over any base, Finite-free local criterion for cohomology and base change).

[F4]

A morphism of finite locally free modules of the same rank which is an isomorphism on every residue-field fibre is an isomorphism; a local basis computation with Nakayama identifies the unit map (A morphism of sheaves is an isomorphism exactly when it is an isomorphism on every stalk, Assuming the Axiom of Choice, Nakayama's lemma).

Proof

technique · direct: compute the degree-zero cohomology fibrewise and apply cohomology and base change

1.1F1F2F3givenalgebra

Work over an affine open $V = Spec R \subseteq S$ , shrinking further so the complex $K$ of [F3] is finite free and concentrated in nonnegative degrees. At any $s \in V$ , the geometric fibre $A_{\overset{ˉ}{s}}$ is smooth and connected, hence integral: regular local rings prevent its finitely many irreducible components from meeting, and connectedness leaves only one. Thus $A_{s}$ is geometrically integral and [F2] gives $H^{0} (A_{s}, O) = κ (s)$ . The actual base-change map $H^{0} (K) \otimes_{R} κ (s) \to H^{0} (K \otimes_{R} κ (s)) = H^{0} (A_{s}, O)$ is surjective, because the global constant section $1$ maps to a basis of its target.

2.1F3step 1.1algebra

Apply the finite-free criterion in [F3] with $q = 0$ to the map just proved surjective. Its preceding map in degree $- 1$ is also surjective, since $K$ has no negative terms. The criterion consequently makes $H^{0} (K)$ finite locally free and gives $H^{0} (K) \otimes_{R} R^{'} ≅ H^{0} (K \otimes_{R} R^{'})$ for every $R$ -algebra $R^{'}$ locally near $s$ . Its residue-field rank is one by step 1.1. Since $s$ was arbitrary, these neighbourhoods cover $S$ , proving that $f_{*} O_{A}$ is invertible and universally compatible with base change.

3.1F3F4step 2.1algebra∎

The unit map $u : O_{S} \to f_{*} O_{A}$ is a morphism of invertible sheaves which over each geometric fibre is an isomorphism (it sends $1$ to the constant function $1$ ); by [F4] it is an isomorphism, and the identity section $e : S \to A$ gives an inverse by pullback of functions, since $e^{*} u = id$ . The same argument applied to $A_{T} \to T$ and to arbitrary base change, including nonreduced $T$ , gives $O_{T} ≅ f_{T, *} O_{A_{T}}$ universally. This argument uses stalkwise and fibrewise isomorphisms supplied by the coherence theorem; it does not infer morphism equality from geometric points.

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Prime to characteristic multiplication is etale

Statement

Assume AC and DC as inherited from the stated suppliers. Let $S$ be a locally Noetherian scheme, let $G \to S$ be a smooth, separated, commutative group scheme of finite type over $S$ (Group schemes over a base scheme, Smooth morphism of schemes), and let $n \geq 1$ be invertible on $S$ (that is, a unit of $O_{S}$ locally). Then the multiplication-by- $n$ endomorphism $[n] : G \to G$ and the kernel $G [n] \to S$ are etale. If moreover $S = Spec R$ for a discrete valuation ring with strict henselization $R^{s h}$ and separably closed residue field $k^{s}$ and $k^{s}$ has characteristic not dividing $n$ , then reduction gives a bijection $G [n] (R^{s h}) \to G [n] (k^{s})$ .

Facts & Assumptions

Given: AC and DC, a locally Noetherian base $S$ , a smooth separated commutative finite-type $S$ -group scheme $G$ , an integer $n$ invertible on $S$ , and, for the last clause, a DVR $R$ with strict henselization $R^{s h}$ and separably closed residue field $k^{s}$ .

[F1]

On a smooth group scheme the tangent space at every point is identified with the translation of the tangent space at the identity, and the differential of a group homomorphism is translation-equivariant; the differential of $[n]$ at the identity is $n$ times the identity because $[n]$ is the sum of $n$ copies of the identity morphism in the group law (Group schemes over a base scheme, Smooth morphism of schemes).

[F2]

On smooth schemes of equal relative dimension the relative Jacobian criterion makes a morphism etale exactly where its differential determinant is invertible; standard smooth presentations and base change give the same over each affine open of the base (Relative Jacobian criterion with its presentation hypothesis, Base change and composition of standard smooth presentations, Differentials of a smooth morphism, Étale equals flat and unramified in finite presentation).

[F3]

Over a strictly henselian local ring with separably closed residue field, reduction is a bijection on points of a separated etale finite-type scheme (Strict henselian etale sections, Unramified residue extensions are finite separable).

Proof

technique · direct: compute the differential of multiplication by $n$, apply the Jacobian criterion, and specialise

1.1F1givenalgebra

At the identity the differential $d [n]_{e}$ is multiplication by $n$ on the tangent space, because $[n]$ is the composite of the $n$ -fold group law and the differential of the group law at the identity is addition; translation identifies the tangent space at every other point with the tangent space at the identity and conjugates $d [n]$ at that point with the corresponding tangent map, so the differential of $[n]$ is everywhere multiplication by the unit $n$ on the locally free tangent sheaf.

2.1F2step 1.1algebra

Since $G$ is smooth over $S$ of constant relative dimension on each connected component and $[n]$ is a morphism between smooth schemes of the same relative dimension, the relative Jacobian criterion in [F2] applies: $[n]$ is etale exactly where the determinant of its differential is a unit, which by step 1.1 holds everywhere since $n$ is invertible on $S$ . Hence $[n]$ is etale. The scheme $G [n]$ is the pullback of $[n]$ along the identity section, so it is etale over $S$ as a base change of an etale morphism; it is separated and of finite type because $G$ is.

3.1F2step 2.1algebra

Negative $n$ is handled by composing with the inversion, which is an isomorphism of $G$ over $S$ . The graph-Jacobian computation of step 2.1 works over each affine open of $S$ , so it does not need $S$ to be a DVR or Noetherian beyond the local Noetherian hypothesis.

4.1F3step 3.1algebra∎

In the DVR case, $G [n] \to Spec R$ is separated etale of finite type, and after the base change to the strictly henselian ring $R^{s h}$ with separably closed residue field $k^{s}$ the general section result [F3] gives that reduction $G [n] (R^{s h}) \to G [n] (k^{s})$ is bijective. This specialization statement is asserted only in this DVR/strictly henselian setting.

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Defect decrease and finite smoothening

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ , and let $R^{s h}$ be a strict henselization.

(a) If $Y \subseteq X_{k}$ is a centre whose $k^{s}$ -points that lift to $R^{s h}$ -sections of $X$ are schematically dense in $Y$ , and $U \subseteq Y$ is a smooth open subscheme on which $Ω_{X / R}$ is locally free, then the $π$ -dilatation of $X$ along $Y$ lowers the positive defect by at least one for every section specializing in $U$ .

(b) For $X$ separated, flat and of finite type over an arbitrary discrete valuation ring with smooth generic fibre, there is a finite sequence of blowups in special-fibre centres, proper and generically isomorphisms, whose smooth locus contains the image of every $R^{s h}$ -section of $X$ .

Facts & Assumptions

Given: AC and DC, a DVR $R$ with uniformizer $π$ , fraction field $K$ , residue field $k$ , a strict henselization $R^{s h}$ , a separated flat finite-type $R$ -scheme $X$ with smooth generic fibre, and a closed centre $Y \subseteq X_{k}$ with schematically dense liftable $k^{s}$ -points.

[F1]

Dilatation charts and the defect computation are Dilatations and defect computation: the $π$ -chart is flat with its universal property, and the defect $δ (a)$ is the torsion length of $a^{*} Ω_{X / R}$ , computed by Jacobian-minor valuations and bounded uniformly on $X (R^{s h})$ .

[F2]

A finitely generated algebra over a field which is injective into a product of copies of the separable closure after evaluation is geometrically reduced, and the smooth locus of a reduced finite-type scheme over a perfect field is dense; separatedness and differential rank control the descent of smoothness through field extensions (Finitely generated extensions of a perfect field are separably generated, Differentials of a separably generated field extension, Field tests for geometric regularity).

[F3]

A coherent sheaf on a reduced finite-type scheme is free on a dense open of every component (Generic freeness over a Noetherian domain).

Proof

technique · direct: reduce the centre to a geometrically reduced dense-smooth object, then run the defect induction on finitely many strata

1.1F2F3givenalgebra

Let $Y$ have schematically dense $k^{s}$ -points. On an affine chart with coordinate ring $B$ , evaluation at the $k^{s}$ -points embeds $B$ into a product of copies of $k^{s}$ ; tensoring with any field extension $l / k$ , every relation involves finitely many coefficients, so the embedding remains injective, and $k^{s} \otimes_{k} l$ is reduced because $k^{s}$ is separable algebraic over $k$ . Hence $B \otimes_{k} l$ is reduced and $Y$ is geometrically reduced. Extending to a perfect closure, the function field of each component is separably generated and the differential rank equals the transcendence degree, so the relative Jacobian criterion produces a smooth neighbourhood of every generic point; smoothness descends through field extensions by [F2]. Therefore the smooth locus of $Y$ is dense and open, and the restriction of $Ω_{X / R}$ to it is free on a dense open by [F3].

2.1F1step 1.1algebra

Shrink around a specialization in $U$ , so $Y = U$ is smooth of dimension $r$ and $Ω_{X / R} ∣_{Y}$ is free of rank $r + n$ . Choose lifts $y_{1}, \dots, y_{r}, z_{1}, \dots, z_{n}$ whose differentials give its basis, with $z_{j}$ vanishing on $Y$ and the $d y_{i}$ mapping to a basis of $Ω_{Y / k}$ . Embed $X$ into affine space with these as initial coordinates. Independent rows of the remaining relation differentials cut out a smooth ambient $Z$ of dimension $r + n$ containing $X$ , with these differentials as a basis. Locally $Y \subset Z$ has ideal $J = (π, z_{1}, \dots, z_{n})$ : its displayed equations define a smooth subscheme of dimension $r$ containing $Y$ , hence agree with $Y$ locally. Put $X = Spec (C / I)$ and $Z = Spec C$ . For $f \in I \subset J$ write $f = π g + \sum z_{j} g_{j}$ . Since the map $Ω_{Z / R} ∣_{Y} \to Ω_{X / R} ∣_{Y}$ identifies the chosen bases, $d f ∣_{Y} = 0$ , so every $g_{j} \in J$ . Thus $f = π g + h$ with $h \in J^{2}$ . On every liftable section through $Y$ , $f (a) = 0$ and $h (a) \in π^{2} R^{s h}$ ; hence $g (a) \in π R^{s h}$ . Schematic density of those specializations gives $g \in J$ . Therefore $I \subseteq J^{2}$ .

3.1F1step 2.1algebra

In the dilatation of $Z$ write $z_{j} = π z_{j}^{'}$ . Since $I \subset J^{2}$ , each relation $f$ becomes $π^{2} f^{'}$ with $f^{'}$ in the saturated ideal of the dilatation of $X$ . Along a lifted section, the Jacobian rows for $f^{'}$ have $y$ -entries $π^{- 2} \partial f / \partial y_{i}$ and $z^{'}$ -entries $π^{- 1} \partial f / \partial z_{j}$ . Choose a maximal-rank minor realizing the old defect in [F1], of size $q = r + n - d$ , where $d$ is the generic relative dimension at the section. The corresponding minor of the divided equations is multiplied by $π^{- 2 a - b}$ , with $a + b = q$ according to its selected coordinate columns. The new defining ideal may have additional generators, so its minimum minor valuation is at most this value: $δ (a^{'}) \leq δ (a) - q$ . If $q = 0$ , the generic closed immersion $X \subset Z$ agrees near the generic section by smoothness and equal dimension; the ideal vanishes locally at the specialization by flatness and schematic density, so the original section is already smooth. Thus positive defect implies $q \geq 1$ , proving (a).

4.1F1F2step 3.1induction

For a set $E$ of nonsmooth $R^{s h}$ -sections, let $Y_{1}$ be the reduced closure of their specializations. It satisfies the liftable density condition by construction. Let $U_{1}$ be its dense smooth open where $Ω_{X / R} ∣_{Y_{1}}$ is locally free, and let $E_{1}$ be the sections specializing there. Repeat with $E ∖ E_{1}$ , obtaining $Y_{2} \subset Y_{1} ∖ U_{1}$ , and continue. The dimensions of the nonempty centres strictly decrease, so this gives a finite partition $E = E_{1} ⊔ \dots ⊔ E_{t}$ with $Y_{i}$ the specialization closure of $E_{i} ⊔ \dots ⊔ E_{t}$ . In particular $Y_{t}$ is permissible for the whole current $E$ : every section of $E$ meeting it belongs to $E_{t}$ and specializes in $U_{t}$ . Blowing up $Y_{t}$ uniquely lifts sections by properness; those through the centre lie in its $π$ -chart, since the pulled-back ideal contains $π$ and is contained in $(π)$ . Their positive defects decrease by step 3.1, while all sections outside the centre are unaffected.

5.1F1step 4.1algebra∎

Use induction first on the uniform maximal defect $D$ from [F1], and within a fixed $D$ on the partition length $t$ . The case $D = 0$ is smooth. Blow up $Y_{t}$ as in step 4.1 and apply the $D - 1$ induction to its lifted subset $E_{t}$ . All resulting centres stay over $Y_{t}$ in the nonsmooth locus, so the other $E_{i}$ are unaffected. Once $E_{t}$ is smooth, the remaining sections have defect at most $D$ and their canonical partition has length $t - 1$ , since the modification is an isomorphism away from $Y_{t}$ ; apply the second induction. This terminates with finitely many permissible special-fibre blowups, proper and generically isomorphisms, smoothing every section of $E$ . Initially take $E$ to be all nonsmooth sections of $X$ . Centres always avoid the smooth locus, so initially smooth sections remain smooth. This proves (b) over every DVR, using no completeness, excellence or perfect-residue hypothesis.

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K-morphisms from smooth models into abelian schemes extend uniquely

Statement

Assume AC and DC. Let $S$ be a Dedekind scheme with function field $K$ , let $A \to S$ be an abelian scheme (Abelian schemes over a base), and let $Z \to S$ be a smooth $S$ -scheme of finite type. Then restriction along the generic fibre $Z_{K} = Z \times_{S} Spec K ↪ Z$ is a bijection ${Hom}_{S} (Z, A) ⟶ {Hom}_{K} (Z_{K}, A_{K}) .$

Facts & Assumptions

Given: AC and DC, a Dedekind scheme $S$ with function field $K$ , an abelian scheme $A \to S$ , a smooth finite-type $S$ -scheme $Z$ , and a $K$ -morphism $u_{K} : Z_{K} \to A_{K}$ .

[F1]

A smooth scheme over the regular Dedekind base is regular, hence normal (Regularity ascends and descends along a flat local homomorphism, Locally standard smooth iff flat with geometrically regular fibres, regular local rings are normal). Its local rings at the generic points of special fibres are discrete valuation rings with fraction field the function field of the component (Height-one localizations of normal Noetherian domains are DVRs, Scheme-theoretic fibre); points and field-valued points correspond as in Field-valued points and local-ring points.

[F2]

A proper morphism satisfies the valuative criterion of properness, so a morphism from the generic point of a valuation ring extends uniquely (Valuative criterion for properness, Abelian schemes over a base).

[F3]

Weil's extension theorem for rational maps into smooth separated group schemes over a regular Noetherian base: a rational map defined in codimension at most one extends uniquely (Weil's extension theorem for rational maps into smooth separated group schemes, S-dense open subschemes and S-rational maps, assuming AC and DC).

[F4]

A morphism into a finitely presented target over a filtered limit of affine schemes descends to a finite stage. For an affine neighbourhood of $ξ$ , its local ring is the filtered limit of the rings of principal neighbourhoods of $ξ$ ; hence an $S$ -morphism $Spec O_{Z, ξ} \to A$ spreads to an open neighbourhood of $ξ$ (Finite-stage descent of finitely presented schemes and their morphisms).

Proof

technique · direct: the valuative criterion at the generic points of the special fibres supplies definedness in codimension one, and Weil's extension theorem extends

1.1F3givenalgebra

Restriction produces a well-defined map ${Hom}_{S} (Z, A) \to {Hom}_{K} (Z_{K}, A_{K})$ , injective because $A$ is separated over $S$ and $Z_{K}$ is schematically dense in the flat $S$ -scheme $Z$ ; so it remains to show surjectivity.

2.1F1F2F4step 1.1construct

For each height-one point $ξ$ of $Z$ lying over a closed point $s \in S$ , [F1] gives a DVR $O_{Z, ξ}$ whose fraction field is the function field of the component of $Z$ containing $ξ$ . The restriction of $u_{K}$ gives a point of $A$ over that fraction field, and properness [F2] extends it to an $S$ -morphism $Spec O_{Z, ξ} \to A$ . By [F4] this map spreads to an open neighbourhood $W_{ξ}$ of $ξ$ in $Z$ . Its restriction to $W_{ξ} \cap Z_{K}$ equals $u_{K}$ , since they agree at the generic point and $A$ is separated. Independently, $u_{K}$ spreads to a morphism on an open neighbourhood $W_{0} \subseteq Z$ containing the whole generic fibre: work locally on a finite-type affine open of the Dedekind base, apply [F4] to its generic localization and the finitely presented smooth source and target, and then glue the resulting restrictions by separatedness. Put $V = W_{0} \cup ⋃_{ξ} W_{ξ}$ . This is an open subscheme containing the generic fibre and every vertical height-one point; every horizontal height-one point lies in the generic fibre. For each closed $s$ , $V_{s}$ contains the generic points of all components of the smooth, hence reduced fibre $Z_{s}$ , so $V$ is $S$ -dense. The local extensions agree with each other and with $u_{K}$ on overlaps: the generic fibre is schematically dense in every open subscheme of the flat $Z$ , and $A$ is separated. Thus they glue to a morphism $V \to A$ , which represents an $S$ -rational map defined at every height-one point.

3.1F3step 2.1algebra∎

The base $S$ is regular Noetherian and $Z$ is smooth over $S$ , so Weil's extension theorem [F3] applies to the $S$ -rational map represented in step 2.1. It extends uniquely to an $S$ -morphism $Z \to A$ restricting to $u_{K}$ . This proves surjectivity and hence the bijection. Uniqueness also follows from separatedness and schematic density of $Z_{K}$ . The argument is applied componentwise; components of a smooth scheme over a Dedekind base are disjoint locally.

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The theorem of the square and the Mumford homomorphism into the Picard group

Statement

Assume the Axiom of Choice. Let $A$ be an abelian variety over a field $k$ (Abelian varieties over a field), let $L$ be an invertible sheaf on $A$ (Picard group of a scheme), and let $t_{a} : A \to A$ denote translation by $a \in A (k^{''})$ for a field extension $k^{''} / k$ . Then the theorem of the square holds: $t_{a + b}^{*} L \otimes L ≅ t_{a}^{*} L \otimes t_{b}^{*} L$ for all $a, b \in A (k^{''})$ , functorially in $k^{''}$ . Consequently the Mumford map $φ_{L} : A (k^{''}) ⟶ Pic (A_{k^{''}}), a ⟼ [t_{a}^{*} L \otimes L^{- 1}],$ is a group homomorphism and lands in the degree-zero part of the rigidified Picard functor (The rigidified relative Picard functor and the dual abelian variety); it is compatible with field extension.

Facts & Assumptions

Given: AC, an abelian variety $A$ over $k$ , an invertible sheaf $L$ on $A$ , a field extension $k^{''} / k$ and points $a, b \in A (k^{''})$ .

[F1]

For an invertible sheaf $L$ on $A$ , the cube theorem gives $m_{123}^{*} L \otimes m_{1}^{*} L \otimes m_{2}^{*} L \otimes m_{3}^{*} L ≅ m_{12}^{*} L \otimes m_{13}^{*} L \otimes m_{23}^{*} L$ on $A^{3}$ , where $m_{I}$ sums the indexed coordinates (The theorem of the cube for an abelian variety). The supplier assumes AC and DC; AC implies DC, since a choice function on the nonempty successor sets of a serial relation defines a sequence by recursion.

[F2]

The Picard group $Pic$ consists of isomorphism classes of invertible sheaves with tensor product, and the rigidified relative Picard functor and its degree-zero part are as defined in The rigidified relative Picard functor and the dual abelian variety, Picard group of a scheme.

Proof

technique · direct: specialise the cube theorem to the standard three maps and read off the homomorphism property

1.1F1givenalgebra

Pull the identity of [F1] back along $A_{k^{''}} \to (A_{k^{''}})^{3}$ , $x \mapsto (x, a, b)$ . Its factors involving $x$ are $t_{a + b}^{*} L$ , $L$ , $t_{a}^{*} L$ and $t_{b}^{*} L$ . The remaining factors are the constant line bundles with fibres $L_{a}$ , $L_{b}$ and $L_{a + b}$ , each a one-dimensional $k^{''}$ -vector space and therefore isomorphic to the trivial line bundle. Removing these constant factors gives $t_{a + b}^{*} L \otimes L ≅ t_{a}^{*} L \otimes t_{b}^{*} L$ . Although trivializations of the constant factors need not be canonical, the resulting equality of Picard classes is canonical and is preserved by field extension.

2.1F1F2step 1.1algebra

The square identity shows that $φ_{L} (a + b) = [t_{a + b}^{*} L \otimes L^{- 1}] = [t_{a}^{*} L \otimes L^{- 1}] [t_{b}^{*} L \otimes L^{- 1}] = φ_{L} (a) φ_{L} (b)$ in $Pic (A_{k^{''}})$ : expand the first factor using the square identity and cancel $L \otimes L^{- 1}$ . Hence $φ_{L}$ is a group homomorphism, and it is natural in $k^{''}$ because the constructions and $L$ are defined over $k$ .

3.1F2step 2.1algebra∎

For degree zero: $φ_{L} (a)$ is represented by the difference of the two line bundles $t_{a}^{*} L$ and $L$ , which occur as fibres of the connected family $L$ over $A$ under the translation family; hence its geometric-fibre restrictions are algebraically equivalent to zero, and $φ_{L}$ lands in the algebraically trivial subfunctor of [F2], i.e. in the degree-zero part of the rigidified Picard functor.

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Fibrewise constant morphisms from an abelian scheme factor through the base

Statement

Assume AC and DC. Let $A \to S$ be an abelian scheme (Abelian schemes over a base), let $T \to S$ be any morphism, write $A_{T} = A \times_{S} T$ with structure morphism $f_{T} : A_{T} \to T$ and unit section $e_{T} : T \to A_{T}$ , and let $Z$ be any scheme. If a morphism $u : A_{T} \to Z$ sends each geometric fibre of $f_{T}$ to a single point, then $u = u \circ e_{T} \circ f_{T}$ as morphisms $A_{T} \to Z$ .

Facts & Assumptions

Given: AC and DC, an abelian scheme $A \to S$ , a morphism $T \to S$ , a scheme $Z$ and a morphism $u : A_{T} \to Z$ which is constant on geometric fibres over $T$ .

[F1]

$O_{T} \to f_{T, *} O_{A_{T}}$ is an isomorphism for every base change, with inverse evaluation along the identity section (Universal structure-sheaf sections of an abelian scheme, assuming AC and DC).

[F2]

A proper morphism has closed image, and properness is stable under base change; the base change $f_{T}$ is proper (Proper morphisms are closed, Properness survives arbitrary base change, Abelian schemes over a base).

[F3]

Morphisms into an affine scheme correspond to ring maps on global sections (Morphisms to an affine scheme and global sections).

Proof

technique · direct: shrink the target to an affine neighbourhood of each fibre image and factor the restriction through the base

1.1F2givenconstruct

Fix $t \in T$ and choose an affine open $W = Spec B \subseteq Z$ containing the image point of the fibre $A_{t}$ ; this is possible because the fibre image is a single point. The complement $Z ∖ W$ is closed, and since $f_{T}$ is proper by [F2] the set $u^{- 1} (Z ∖ W)$ has closed image in $T$ ; by construction that image misses $t$ . Choose an affine neighbourhood $V = Spec R$ of $t$ disjoint from the image; then $f_{T}^{- 1} (D)$ for $D = T ∖ V$ is closed in $A_{T}$ and disjoint from $A_{V}$ , so $u ∣_{A_{V}}$ lands in $W$ . Thus over an affine neighbourhood of every point the map factors through an affine target.

2.1F1F3step 1.1algebra

On $A_{V}$ the structure morphism $f_{V} : A_{V} \to V$ is proper and the restriction $u_{V} : A_{V} \to W = Spec B$ corresponds by [F3] to a ring map $B \to Γ (A_{V}, O_{A_{V}})$ . The universal-sections lemma [F1] identifies $Γ (A_{V}, O_{A_{V}}) = Γ (V, O_{V}) = R$ , so this ring map factors through $R$ and defines a morphism $g_{V} : V \to W = Spec B$ with $u ∣_{A_{V}} = g_{V} \circ f_{V}$ . Evaluating along the unit section gives $u ∣_{A_{V}} \circ e_{V} = g_{V}$ , so $u ∣_{A_{V}} = u ∣_{A_{V}} \circ e_{V} \circ f_{V}$ .

3.1F1step 2.1algebra∎

The local factorizations of step 2.1 agree on overlaps: on $V_{1} \cap V_{2}$ both $g_{V_{1}}$ and $g_{V_{2}}$ equal $u \circ e$ evaluated there, because $f$ restricted to the unit section is an isomorphism onto the base; hence they glue to a morphism $g : T \to Z$ with $u = g \circ f_{T}$ , and $g = u \circ e_{T}$ by the same evaluation. Therefore $u = u \circ e_{T} \circ f_{T}$ . This controls nilpotents: the factorization is an identity of morphisms, not merely of geometric points.

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Field prime to characteristic torsion and Tate module

Statement

Assume AC and DC as inherited from the stated suppliers. Let $A$ be an abelian variety of dimension $g$ over a field $F$ and let $ℓ \neq char F$ be prime. Then for every $ν \geq 1$ :

(a) $A [ℓ^{ν}] (F^{s e p}) ≅ (Z / ℓ^{ν} Z)^{2 g}$ , and multiplication by $ℓ$ induces surjective transition maps $A [ℓ^{ν + 1}] (F^{s e p}) \to A [ℓ^{ν}] (F^{s e p})$ ;

(b) $T_{ℓ} (A) ≅ Z_{ℓ}^{2 g}$ (Prime-to-residue-characteristic Tate modules and inertia), and the natural projections give $T_{ℓ} (A) / ℓ^{ν} T_{ℓ} (A) ≅ A [ℓ^{ν}] (F^{s e p})$ ;

(c) an automorphism of $F^{s e p}$ over $F$ (in particular an inertia group element) acts trivially on $T_{ℓ} (A)$ if and only if it acts trivially on every $A [ℓ^{ν}] (F^{s e p})$ .

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ of dimension $g$ over a field $F$ , a prime $ℓ \neq char F$ , and a separable closure $F^{s e p}$ .

[F1]

Multiplication by $n$ on an abelian variety is finite, flat and surjective of degree $n^{2 g}$ in the sense that $A [n]$ is finite locally free of rank $n^{2 g}$ ; when $n$ is invertible in the field, the group $A [n] (\overset{ˉ}{F})$ has $n^{2 g}$ elements (Nonzero multiplication on an abelian variety is finite and faithfully flat, assuming AC and DC).

[F2]

For $ℓ$ invertible, $[ℓ] : A \to A$ and $A [ℓ^{ν}] \to Spec F$ are etale, so the geometric points of $A [ℓ^{ν}]$ are the separable ones, and reduction is injective on torsion over strictly henselian bases (Prime to characteristic multiplication is etale).

[F3]

A finite abelian $ℓ$ -group with $ℓ^{2 g ν}$ elements killed by $ℓ^{ν}$ , whose $ℓ$ -torsion has $ℓ^{2 g}$ elements is isomorphic to $(Z / ℓ^{ν} Z)^{2 g}$ (Fundamental theorem of finite abelian groups: elementary-divisor form); separable closures exist and are unique up to $F$ -isomorphism (Assuming Choice, separable closures exist and are base-isomorphic).

Proof

technique · direct: count torsion, classify the finite abelian groups, and take the inverse limit with its quotient description

1.1F1F2F3givenalgebra

By [F1] $A [ℓ^{ν}]$ is finite locally free of rank $ℓ^{2 g ν}$ ; by [F2] it is etale over $F$ , so its geometric points are separable and $∣ A [ℓ^{ν}] (F^{s e p}) ∣ = ℓ^{2 g ν}$ . In particular $A [ℓ] (F^{s e p})$ has $ℓ^{2 g}$ elements, and $H = A [ℓ^{ν}] (F^{s e p})$ is a finite abelian $ℓ$ -group whose $ℓ$ -torsion has $ℓ^{2 g}$ elements; by the elementary divisor classification [F3], $H ≅ (Z / ℓ^{ν} Z)^{2 g}$ .

2.1F1F3step 1.1construct

Multiplication by $ℓ$ maps $H_{ν + 1} = A [ℓ^{ν + 1}] (F^{s e p})$ into $H_{ν}$ with kernel $H_{1}$ of order $ℓ^{2 g}$ . The order computation in step 1.1 makes its image have order $ℓ^{2 g ν}$ , so it is surjective. Choose a basis of $H_{1}$ and recursively lift each basis vector through these maps. The lifted vectors form a basis of $H_{ν + 1}$ over $Z / ℓ^{ν + 1} Z$ : a relation, after applying $ℓ$ , has all coefficients divisible by $ℓ^{ν}$ by the basis property in $H_{ν}$ ; multiplying the lifted vectors by $ℓ^{ν}$ gives the original basis of $H_{1}$ , so the remaining coefficients are zero modulo $ℓ$ . Independence and the equal orders then give generation. DC (and hence the assumed AC) permits the countable recursive choice of compatible bases. These compatible bases identify the inverse system with the reductions of $(Z_{ℓ})^{2 g}$ , and hence identify its inverse limit with that module.

3.1F3step 2.1algebra

The projections $T_{ℓ} (A) \to H_{ν}$ are surjective by the compatible-basis construction. Their kernel is $ℓ^{ν} T_{ℓ} (A)$ . Indeed the inclusion from right to left follows because $H_{ν}$ is killed by $ℓ^{ν}$ . Conversely, for a compatible sequence $(a_{r})$ with $a_{ν} = 0$ , put $b_{r} = a_{r + ν}$ . Compatibility gives $ℓ^{r} b_{r} = a_{ν} = 0$ , so $b_{r} \in H_{r}$ , and $ℓ b_{r + 1} = b_{r}$ , so $(b_{r}) \in T_{ℓ} (A)$ . Also $ℓ^{ν} b_{r} = a_{r}$ , giving the reverse inclusion. This proves $T_{ℓ} (A) / ℓ^{ν} T_{ℓ} (A) ≅ H_{ν}$ .

4.1F2step 3.1algebra∎

For (c), an element $σ$ of ${Aut}_{F} (F^{s e p})$ acts on $T_{ℓ} (A)$ coordinatewise and on each $A [ℓ^{ν}] (F^{s e p})$ by functoriality; the quotient identifications of step 3.1 are $σ$ -equivariant, so $σ$ acts trivially on $T_{ℓ} (A)$ if and only if it acts trivially on each quotient, i.e. on every finite torsion group. This is an elementary group argument on top of the multiplication supplier; no $H^{1}$ duality statement is asserted, and the choice assumptions of [F1] persist.

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Projective weak models and rational mapping

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ , residue field $k$ , and strict henselization $R^{s h}$ , and let $A$ be an abelian variety over $K$ of dimension $g$ .

(a) There is a smooth separated finite-type $R$ -model $V$ of $A$ with $V (R^{s h}) = A (K^{s h})$ ; no excellence, bounded-model theorem or flattening hypothesis is used.

(b) A weak model collection for $A$ receives every generic rational map from a smooth $R$ -scheme $Z$ with irreducible special fibre as an $R$ -rational map into one of its members.

Facts & Assumptions

Given: AC and DC, a DVR $R$ with fraction field $K$ , residue field $k$ and strict henselization $R^{s h}$ , an abelian variety $A / K$ of dimension $g$ , and a weak model collection for $A$ .

[F1]

An abelian variety over $K$ is projective, and the schematic closure of $A$ in a projective $R$ -space is proper of finite type with generic fibre $A$ ; proper morphisms satisfy the valuative criterion, and the finite permissible smoothening produces a finite sequence of generically identical special-fibre blowups whose smooth locus contains every $R^{s h}$ -section (Every abelian variety over a field is projective, Valuative criterion for properness, Defect decrease and finite smoothening, Schematic closure and agreement on a dense open, Strict henselization of a DVR and smooth sections).

[F2]

Over a Noetherian base $B$ , a fibre-dense open of a smooth $B$ -scheme is schematically dense. A prime filtration of $B$ remains a filtration after tensoring with a flat smooth $B$ -algebra $C$ , with factors $C / p C$ . Each factor is flat over the domain $B / p$ and injects into its generic fibre, which is geometrically regular and reduced. Fibre density makes restriction injective on that generic fibre, hence on each factor, and induction makes restriction injective on $C$ . This proves density without asserting that the associated primes of $C$ are minimal. Flat tensor products preserve finite kernels and equalizers (Finite modules over Noetherian rings admit prime filtrations, Filtered colimits of abelian groups are exact, S-dense open subschemes and S-rational maps).

[F3]

Domains of $S$ -rational maps descend along faithfully flat smooth maps and commute with flat base change, the graph closure being computed by finite kernels of restriction maps; morphisms into a separated target that agree on a schematically dense open are equal (S-dense open subschemes and S-rational maps, An S-rational map defined after a faithfully flat smooth base change is defined, Scheme morphisms satisfy fppf descent, Agreement on a schematically dense open); images of finitely presented morphisms are constructible (Constructible images for finite-presentation affine maps).

Proof

technique · direct: close $A$ projectively, smoothen, then construct the rational map by graph closures and constructibility, with the density and descent facts of [F2] and [F3]

1.1F1givenconstruct

By [F1] choose a projective embedding of $A$ over $K$ , let $X$ be the schematic closure of $A$ in the corresponding projective $R$ -space, and note that $X$ is proper of finite type over $R$ with generic fibre $A$ . Its chart rings are $π$ -torsion-free, since schematic closure contracts the generic ideal, so $X$ is flat over the DVR. For every $a \in A (K^{s h}) = X (K^{s h})$ , the valuative criterion of properness extends $a$ uniquely to an $R^{s h}$ -point of $X$ ; applying the finite permissible smoothening theorem [F1] to $X$ produces a finite sequence of special-fibre blowups, proper and generically identical, whose smooth locus contains every such section. The smooth locus $V$ of the resulting model is therefore a smooth separated finite-type $R$ -model of $A$ with $V (R^{s h}) = A (K^{s h})$ , proving (a).

2.1F2step 1.1algebra

For a fibre-dense open $U$ in a smooth scheme over a Noetherian base, apply the filtration argument of [F2] on an affine source chart. More explicitly, a function zero on $U$ maps to zero in the last factor's generic fibre, because that fibre is reduced and $U$ meets every irreducible component. Flatness over the domain $B / p$ injects the factor into its generic fibre. The function therefore lies in the previous filtration submodule, where it still restricts to zero; induction through the finite filtration gives zero. Thus restriction is injective. The same proof applies after any Noetherian base change for which the source remains smooth and $U$ remains fibre-dense, in particular the DVR localizations and strict henselizations used here. Schematic graph closures commute with flat base change because restriction kernels do: compute restriction with a finite affine cover of $U$ , use its finite equalizer, and tensor with a flat algebra. No claim that a smooth algebra over an arbitrary Noetherian base has only minimal associated primes is used.

3.1F2F3step 2.1algebra

The graph closure of a rational map is computed on a finite affine cover by kernels of restriction maps, so it commutes with flat base change and the domain of definition is the open where the graph projection is an isomorphism; if a faithfully flat pullback of that projection is an isomorphism, affine ring descent gives the isomorphism before pullback by [F3]. Hence domains descend along faithfully flat source maps and commute with flat base changes, and representatives agreeing on a schematically dense open of a separated target are equal. This is the BLR relative-rational-map statement of Chapter 2, Section 5 in the form used below.

4.1F2F3step 3.1construct

For (b), shrink $Z$ fibre-densely so the generic rational map is defined on its whole generic fibre: the closure of the excluded proper generic closed set is nowhere dense in the smooth irreducible special fibre by the DVR dimension argument. Let $Γ_{i} \subseteq Z \times_{R} V_{i}$ be its schematic graph closure for each of the finitely many members of the weak collection, with projection $p_{i}$ to $Z$ . After passing to $R^{s h}$ , every rational special point of $Z$ lifts to a section by [F1]; its generic image extends to a section of some $V_{i}$ by weakness, and the paired section lies in $Γ_{i}$ by schematic closure. These special points are dense. Constructibility [F3] and finiteness of the collection therefore force some image $p_{i} (Γ_{i})$ to contain the generic point $η$ of the original special fibre (this can be checked after the faithfully flat strict-henselian extension). Choose $q \in Γ_{i}$ over $η$ . The graph is flat and generically isomorphic to $Z$ , so $O_{Γ_{i}, q}$ and the DVR $O_{Z, η}$ lie in the same function field. The former is a local ring dominating that DVR; a proper overring of a DVR in its fraction field inverts the uniformizer and cannot dominate it, so the two rings coincide. Finite-type affine presentations and clearing denominators spread this stalk equality to an isomorphism on neighbourhoods of $q$ and $η$ . Composing its inverse with the other graph projection extends the generic map on an $R$ -dense open of $Z$ , proving (b).

5.1F2F3step 4.1algebra∎

For (c), put $R^{'} = O_{Z, η}$ and $K^{'} = Frac R^{'}$ . A point of $A (K^{' s h})$ uses only finitely many coordinates and relations, so it is defined over the fraction field of a pointed local etale neighbourhood of $R^{'}$ . Spread that neighbourhood to an etale scheme $Z^{'} \to Z$ near its special generic point $η^{'}$ , and spread the point to a generic rational map $Z_{K}^{'} ⇢ A$ . Shrink to the open consisting of the generic fibre and the special component containing $η^{'}$ , so (b) supplies an $R$ -rational map into some $V_{i}$ defined at $η^{'}$ . Localizing and then passing to $R^{' s h}$ extends the given point to an $R^{' s h}$ -section of $(V_{i})_{R^{'}}$ . Thus the base-changed finite collection is weak. The extension $K^{'} / K$ may have transcendence degree; no finite-separable identification of the two fraction fields is used.

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Rigidification and effective descent of line bundles

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be an abelian variety over a field $k$ with identity $e$ (Abelian varieties over a field) and let $T$ be a $k$ -scheme, with $A_{T} = A \times_{k} T$ , unit section $e_{T}$ , and projection $p_{T} : A_{T} \to T$ . Then the rigidified line-bundle functor of The rigidified relative Picard functor and the dual abelian variety is an fppf sheaf on all $k$ -schemes, rigidified line bundles have no nontrivial automorphisms, and the normalization $L \mapsto L \otimes p_{T}^{*} e_{T}^{*} L^{- 1}$ identifies the rigidified classes over $T$ with $Pic (A_{T}) / p_{T}^{*} Pic (T)$ .

Facts & Assumptions

Given: AC and DC, an abelian variety $A / k$ with identity $e$ , a $k$ -scheme $T$ , and the base-changed abelian scheme $A_{T} \to T$ .

[F1]

For every $T$ the unit map $O_{T} \to p_{T, *} O_{A_{T}}$ is an isomorphism with inverse evaluation along $e_{T}$ ; in particular every global function comes from the test base (Universal structure-sheaf sections of an abelian scheme, assuming AC and DC).

[F2]

Global sections are compatible with flat field base change, so the computation of $H^{0}$ may be done after extending the base field (Global sections commute with extension of scalars over a field); faithfully flat descent of modules and algebras is effective (Faithfully flat descent of modules and algebras is effective).

Proof

technique · direct: normalize to remove constants, observe automorphism-freeness, and descend along an fppf cover

1.1F1F2givenalgebra

For an affine test $T = Spec R$ the universal-sections statement [F1] gives $Γ (A_{T}, O) = R$ compatibly with base change; on a finite affine Cech cover of $A_{T}$ the cohomology complex computing $H^{0}$ is obtained by tensoring the field cohomology complex, whose $H^{0}$ is $k$ , and hence has $H^{0} = R$ . It follows that the normalization $L \mapsto L \otimes p_{T}^{*} e_{T}^{*} L^{- 1}$ is well defined on isomorphism classes and identifies rigidified classes with $Pic (A_{T}) / p_{T}^{*} Pic (T)$ : tensoring by constants is exactly the ambiguity removed by the trivialisation along $e_{T}$ .

2.1F1step 1.1algebra

A rigidified line bundle has no nontrivial automorphism: an automorphism of $(L, α)$ is a unit of $O_{T}$ acting on $L$ , and compatibility with the rigidification forces it to restrict to $1$ along $e_{T}$ ; since $e_{T}$ is a section, the unit is $1$ . Consequently isomorphism data on overlaps of an fppf cover are unique and therefore automatically satisfy the cocycle condition.

3.1F2step 2.1algebra

Let $T^{'} \to T$ be an fppf cover and suppose a rigidified line bundle is given on $A_{T^{'}}$ together with an isomorphism of its two pullbacks to $A_{T^{'} \times_{T} T^{'}}$ ; by step 2.1 this isomorphism is unique and satisfies the cocycle condition, so the usual effective descent for invertible modules [F2] produces an invertible sheaf on $A_{T}$ ; the rigidification descends because it is a morphism whose pullbacks agree. Hence the rigidified functor is already an fppf sheaf, without invoking representability, and the identification of step 1.1 is compatible with the sheaf structure.

4.1F2step 3.1algebra∎

For general $k$ one first verifies the assertions over an algebraic closure using the field-compatibility of global sections in [F2] and then descends the resulting identifications along the faithfully flat field extension; the rigidification data are defined over $k$ and descend by [F2]. No representability of the Picard functor is used anywhere.

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Fibres of abelian schemes and unit-preserving morphisms

Statement

Assume AC and DC. Let $S$ be a scheme, let $A \to S$ and $B \to S$ be abelian schemes of relative dimensions $g_{A}, g_{B}$ (Abelian schemes over a base), and let $s \in S$ . Then:

(a) the fibre $A_{s}$ is an abelian variety of dimension $g_{A}$ over $κ (s)$ ;

(b) the multiplication of $A$ is commutative and the inversion is the morphism $- 1_{A} : A \to A$ ;

(d) consequently, on a connected base, any two abelian-scheme group structures on the same smooth proper $S$ -scheme with the same unit section coincide.

Facts & Assumptions

Given: AC and DC, abelian schemes $A \to S$ , $B \to S$ and a point $s \in S$ .

[F1]

An abelian scheme has smooth proper connected geometric fibres of constant dimension (Abelian schemes over a base); the fibre over $s$ is the base change to $κ (s)$ (Scheme-theoretic fibre, Field-valued points and local-ring points).

[F2]

Every abelian variety over a field is commutative, and a pointed morphism from a smooth geometrically integral group variety to an abelian variety is a homomorphism (A proper geometrically connected group variety is commutative, Pointed morphisms from smooth geometrically integral groups to abelian varieties are homomorphisms, Abelian varieties over a field).

[F3]

A morphism of abelian schemes over $S$ which is constant on every geometric fibre factors through the base (Fibrewise constant morphisms from an abelian scheme factor through the base).

Proof

technique · direct: the field-level statements applied fibrewise, then the rigidity factorization to pass to morphisms

1.1F1givenalgebra

The fibre $A_{s} = A \times_{S} Spec κ (s)$ is smooth, proper and geometrically connected of dimension $g_{A}$ over $κ (s)$ by [F1], hence an abelian variety of dimension $g_{A}$ ; this is (a).

2.1F2F3step 1.1algebra

For commutativity, let $c : A \times_{S} A \to A$ be the commutator morphism $c (a, b) = a b a^{- 1} b^{- 1}$ , using the group law; it sends the unit sections to the unit. For each geometric point $\overset{ˉ}{t}$ of the base, the fibre of $A \times_{S} A \to S$ over $\overset{ˉ}{t}$ is $A_{\overset{ˉ}{t}} \times_{\overset{ˉ}{t}} A_{\overset{ˉ}{t}}$ , and by the field-level commutativity [F2] the commutator is constant, equal to the identity, on each geometric fibre of the second projection; by [F3] applied to the base change $A_{A} \to A$ (second projection), $c$ factors through the base, and evaluating at the first unit section gives $c = 1$ , hence $a b = b a$ as morphisms. This proves the first claim of (b), including nilpotents.

3.1F1step 2.1algebra

For the inverse: $m ({id}_{A}, - 1_{A})$ and $e \circ f$ agree on the closed subscheme $A$ by the group axioms, so the inverse is $- 1_{A}$ as defined; this is the second claim of (b).

4.1F2F3step 2.1algebra∎

For (c), let $u : A \to B$ satisfy $u \circ e_{A} = e_{B}$ and consider the defect morphism $d (a, b) = u (a + b) - u (a) - u (b)$ on $A \times_{S} A$ , using the group law of $B$ . On each geometric fibre of the first projection, the field-level pointed-morphism theorem [F2] makes $d$ constant, equal to $0$ ; by [F3] it factors through the base and evaluation at $a = e_{A}$ gives $d \equiv 0$ , so $u$ is additive; compatibility with the unit is assumed, so $u$ is a homomorphism of $S$ -group schemes. For (d), two group structures on the same $S$ -scheme with the same unit section have an identity morphism which preserves the unit, hence is a homomorphism by (c), and being an isomorphism of underlying schemes it identifies the two structures.

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Polarizations and the Mumford isogeny attached to an ample line bundle

Definition

Assume AC for the cited square theorem. Let $A$ be an abelian variety over a field $k$ (Abelian varieties over a field), and suppose a dual abelian variety $A^{\lor}$ with its universal normalized Poincare bundle $P$ on $A \times_{k} A^{\lor}$ has been supplied (The rigidified relative Picard functor and the dual abelian variety).

For an invertible sheaf $L$ on $A$ the Mumford morphism $φ_{L} : A \to A^{\lor}$ is the homomorphism of The theorem of the square and the Mumford homomorphism into the Picard group; it is represented by the family $Λ (L) = m^{*} L \otimes p_{1}^{*} L^{- 1} \otimes p_{2}^{*} L^{- 1} \otimes π^{*} e^{*} L$ on $A \times_{k} A$ , rigidified along both identity factors, where $m : A \times_{k} A \to A$ is the group law, $p_{1}, p_{2}$ are the projections, $π : A \times_{k} A \to Spec k$ is the structure morphism for the constant factor, and $e$ is the identity section; in particular $({id}_{A} \times φ_{L})^{*} P ≅ Λ (L)$ , and the map used here has domain $A \times_{k} A$ .

A polarization of $A$ is a homomorphism $λ : A \to A^{\lor}$ such that, after extension of scalars to an algebraic closure of $k$ , there exists an ample invertible sheaf $L$ on $A_{\overset{ˉ}{k}}$ (Absolute ampleness by affine section opens) with $λ_{\overset{ˉ}{k}} = φ_{L}$ . A principal polarization is a polarization of degree one, where the degree of a polarization is the finite locally free rank of the associated isogeny, once $λ$ is known to be an isogeny.

This definition is conditional on the supply of the dual and the Poincare bundle; symmetry $λ = λ^{\lor}$ under the bidual identification, finiteness and the isogeny property, existence of the dual, and existence and square degree of polarizations are conclusions to be proved in the subsequent commissioned theorem and are not assumed as existence assertions here. Defining these conditional terms does not create a dependency from this definition back to that theorem.

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Special fibre torsion growth detects properness

Statement

Assume AC and DC as inherited from the stated suppliers. Let $k$ be a field and let $G$ be a smooth commutative finite-type $k$ -group scheme of dimension $g$ . Fix a prime $ℓ \neq char k$ . If $∣ G [ℓ^{ν}] (\overset{ˉ}{k}) ∣ = ℓ^{2 g ν}$ for every $ν \geq 1$ , then the identity component $G^{0}$ is an abelian variety over $k$ (in particular $G^{0}$ is proper).

Facts & Assumptions

Given: AC and DC, a field $k$ , a smooth commutative finite-type $k$ -group scheme $G$ of dimension $g$ , and a prime $ℓ \neq char k$ with $∣ G [ℓ^{ν}] (\overset{ˉ}{k}) ∣ = ℓ^{2 g ν}$ for all $ν$ .

[F1]

Over the algebraic closure, the identity component of a smooth connected commutative group variety is an extension of an abelian variety by a smooth connected affine group (Barsotti-Chevalley over a perfect field: unique smooth affine normal subgroup); the affine part has a prime-to-characteristic torsion bound $∣ N [ℓ^{ν}] ∣ \leq ℓ^{ν \dim N}$ (Prime-to-characteristic torsion bound for affine commutative groups, assuming AC and DC).

[F2]

On an abelian variety of dimension $b$ , $∣ B [ℓ^{ν}] (\overset{ˉ}{k}) ∣ = ℓ^{2 b ν}$ (Field prime to characteristic torsion and Tate module). For a finite-type group scheme over a field, the identity is a closed rational point and the diagonal is the inverse image of the identity under $(x, y) \mapsto x y^{- 1}$ , so the group scheme is separated (Group schemes over a base scheme); consequently prime-to-characteristic multiplication has etale finite-type kernel by Prime to characteristic multiplication is etale, and over a field that kernel is finite etale. Thus passage from $k^{s e p}$ to $\overset{ˉ}{k}$ changes no torsion points. Geometric properness descends through field extensions (Properness over a field can be checked after field extension).

Proof

technique · direct: split the identity component into abelian and affine parts, bound torsion componentwise, and let $\nu\to\infty$

1.1F1F2givenalgebra

Over $\overset{ˉ}{k}$ write $0 \to N \to G_{\overset{ˉ}{k}}^{0} \to B \to 0$ with $N$ smooth connected affine of dimension $a$ and $B$ an abelian variety of dimension $b$ , so that $g = a + b$ ; this is the Barsotti-Chevalley decomposition of [F1]. The torsion of $G_{\overset{ˉ}{k}}^{0} [ℓ^{ν}]$ maps to $B [ℓ^{ν}]$ with fibres that are torsors under $N [ℓ^{ν}]$ , of cardinal at most $∣ N [ℓ^{ν}] ∣ \leq ℓ^{a ν}$ by [F1], while $∣ B [ℓ^{ν}] ∣ = ℓ^{2 b ν}$ by [F2]. Hence $∣ G^{0} [ℓ^{ν}] (\overset{ˉ}{k}) ∣ \leq ℓ^{(2 b + a) ν} = ℓ^{(2 g - a) ν}$ .

2.1F1step 1.1algebra

The group $G$ has finitely many connected components, say $C$ of them, and translation by a torsion point in a component embeds that component's torsion into $G^{0} [ℓ^{ν}]$ (subtracting a torsion point identifies the component with $G^{0}$ and preserves torsion), so $∣ G [ℓ^{ν}] (\overset{ˉ}{k}) ∣ \leq C \cdot ∣ G^{0} [ℓ^{ν}] (\overset{ˉ}{k}) ∣ \leq C ℓ^{(2 g - a) ν}$ . Comparing with the hypothesis $ℓ^{2 g ν}$ gives $ℓ^{a ν} \leq C$ for all $ν \geq 1$ , which forces $a = 0$ .

3.1F1F2step 2.1algebra∎

Therefore $N$ is trivial and $G_{\overset{ˉ}{k}}^{0} = B$ is an abelian variety. By [F2], each $G [ℓ^{ν}]$ is finite etale, so passage from $k^{s e p}$ to $\overset{ˉ}{k}$ changes no torsion points; the count is therefore already available over $k^{s e p}$ . Geometric properness of $G^{0}$ descends from $\overset{ˉ}{k}$ to $k$ by [F2], so $G^{0}$ is proper over $k$ and, being smooth connected commutative and proper, is an abelian variety over $k$ . This bound suffices for the arithmetic applications and avoids asserting a stronger exact exponent.

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Invariant volume and finite minimal classes

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ , uniformizer $π$ , residue field $k$ . For the model assertions fix an abelian variety $A / K$ and a nonzero invariant top form $ω$ on $A$ ; models are smooth separated finite-type $R$ -models of this fixed $A$ with nonempty irreducible special fibre. Their order is the valuation of $ω$ at the special generic point. Two models are equivalent when they have isomorphic $R$ -dense opens inducing the identity on $A$ .

(a) A smooth $d$ -dimensional $R$ -group scheme has a nowhere-vanishing invariant top form; on a smooth model $X$ with irreducible special fibre, $π^{- ord}$ times the generic form extends to a generator, where $ord$ measures the vanishing of the normalized form.

(b) An $R$ -rational, generically identical map $φ : X ⇢ Y$ between smooth models with irreducible special fibre satisfies $ord X \geq ord Y$ , with equality if and only if $φ$ is etale on its domain; in particular equal orders imply $φ$ is an open immersion on its domain.

(c) Orders have a finite minimum and there are finitely many equivalence classes of minimal models; minimal representatives remain minimal and cover all minimal classes after the base change $R \to O_{Z, η}$ at a generic point of a special fibre, after splitting special components.

Facts & Assumptions

Given: AC and DC, a DVR $R$ with uniformizer $π$ , fraction field $K$ and residue field $k$ , and a smooth finite-type $R$ -group scheme (or a smooth model with irreducible special fibre).

[F1]

The cotangent space at the identity of a smooth group scheme is locally free of rank the relative dimension and is translation-invariant, so its top exterior power trivializes the sheaf of invariant $d$ -forms (Differentials of a smooth morphism, Locally standard smooth iff flat with geometrically regular fibres); Hartogs extension in codimension one and the pure-codimension-one support of zeros of sections are available on regular total spaces (A normal Noetherian domain is the intersection of its height-one localizations, Regular local rings are unique factorization domains, Rational sections of line bundles are Cartier divisors).

[F2]

A morphism between smooth schemes of equal relative dimension is etale exactly where its relative differential determinant is invertible, and a quasi-finite birational separated morphism onto a normal target with reduced source is an open immersion (Etale morphisms are the formally etale morphisms locally of finite presentation, Scheme Zariski Main factorization for separated quasi-finite morphisms, Regularity ascends and descends along a flat local homomorphism).

[F3]

A weak model collection receives every generic rational map from a smooth $R$ -scheme with irreducible special fibre, and weak models remain weak after base change to a special-fibre generic local ring (Projective weak models and rational mapping).

Proof

technique · direct: trivialize the invariant top form by translation, compare orders along rational maps, and use the weak collection to bound the minimum

1.1F1givenconstruct

On a smooth $d$ -dimensional group scheme the cotangent module at the identity is free of rank $d$ over the DVR. Choose a generator of its top exterior power and translate it by the group law; the translation trivialization gives an invariant top form which generates at every point, hence is nowhere vanishing. For a smooth model with irreducible special fibre, let $ord$ be the valuation of its nowhere-vanishing generic invariant form $ω$ at the special generic point, and write $ω = π^{ord} ω_{0}$ there. The normalized form $π^{- ord} ω$ has neither zeros nor poles along the special component, and none along horizontal prime divisors since the generic invariant form is nowhere zero. Hartogs therefore extends it over the regular total space. Its zero locus would have a prime-divisor component by [F1], but no such divisor is available, so it is a generator everywhere. This proves (a). For an abelian variety, left and right invariance agree by commutativity, so the general bounded modular-character argument is unnecessary.

2.1F1F2step 1.1algebra

Let $φ : X ⇢ Y$ be $R$ -rational and generically identical between smooth models with irreducible special fibres. Pulling back the normalized invariant form of $Y$ along $φ$ gives a rational form on $X$ whose scalar coefficient relative to the normalized form of $X$ is $π^{ord X - ord Y}$ times a unit; invariance under the generic identification and the divisor computation of step 1.1 give $ord X \geq ord Y$ . If equality holds, the relative differential determinant of $φ$ is a unit on its domain, so $φ$ is etale there by [F2]; a generically identical etale separated morphism with reduced source onto a normal target is an open immersion by the Zariski Main Theorem argument in [F2]. This proves (b).

3.1F1F2F3step 2.1algebra∎

Split a finite weak model collection into its finitely many models with irreducible special fibre. By [F3], every smooth model $X$ with irreducible special fibre has an $R$ -rational map, generically the identity, into some member $V_{i}$ . Step 2.1 gives $ord (X) \geq ord (V_{i})$ , so the finite minimum of the collection's orders is a lower bound for all model orders. Conversely each $V_{i}$ is itself a model, so a member attaining that minimum is minimal among all models. A minimal $X$ maps into a $V_{i}$ with the same order; step 2.1 makes that map an open immersion on a fibre-dense domain, giving equivalence with one of the finitely many minimal collection members. After the smooth-DVR base change $R \to O_{Z, η}$ , [F3] keeps the collection weak, and the uniformizer and the normalized volume orders of its split components are unchanged. The same lower-bound argument therefore preserves the minimum and covers all minimal classes by the base-changed representatives, proving (c).

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Cube-derived square over DVR

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ , and let $H$ be a smooth separated finite-type $R$ -group scheme whose generic fibre is abelian, with identity component $H^{0} = H_{K} \cup (H_{k})^{0}$ as in The identity model of a smooth group with abelian generic fibre.

(a) For an abelian variety $A / K$ and every invertible sheaf $L$ on $A$ , the square obstruction on $A \times_{K} A \times_{K} A$ is pulled back from the first two factors.

(b) Every invertible sheaf on $H$ satisfies the theorem of the square for the translation action of $H^{0}$ .

No Picard representability, dual abelian variety, Chevalley decomposition or Raynaud theorem is used.

Facts & Assumptions

Given: AC and DC, a DVR $R$ with fraction field $K$ and residue field $k$ , a smooth separated finite-type $R$ -group scheme $H$ with abelian generic fibre, and an invertible sheaf $L$ on $H$ .

[F1]

The theorem of the cube: for an abelian variety $A$ over a field and every invertible sheaf on $A \times A \times A$ expressed in the standard way, the alternating product of its pullbacks under the partial sums is trivial (The theorem of the cube for an abelian variety); the field-level theorem of the square is its two-variable consequence (The theorem of the square and the Mumford homomorphism into the Picard group).

[F2]

The identity component $H^{0}$ is an open subgroup scheme with geometrically connected and geometrically irreducible fibres, and its orbits on geometric fibres are the connected components (The identity model of a smooth group with abelian generic fibre).

[F3]

Smooth total spaces over the DVR are regular, regular local rings are UFDs, so Weil divisors are locally Cartier and generic divisors extend by closing their prime supports; the Cartier divisor/rational section correspondence is available, and scheme Hartogs extends sections defined in codimension one (Regularity ascends and descends along a flat local homomorphism, Locally standard smooth iff flat with geometrically regular fibres, Regular local rings are unique factorization domains, Rational sections of line bundles are Cartier divisors, A normal Noetherian domain is the intersection of its height-one localizations, Scheme Zariski Main factorization for separated quasi-finite morphisms).

Proof

technique · direct: rewrite the cube identity as a pullback identity, then extend the generic square by divisor closure and absorb the residual constant factor

1.1F1givenalgebra

Write the cube identity for $L$ in the standard form $L (x + y + z) \otimes L (z) ≅ L (x + z) \otimes L (y + z) \otimes L (x + y) \otimes L (x)^{- 1} \otimes L (y)^{- 1},$ read as an isomorphism of pullbacks on $A \times_{K} A \times_{K} A$ modulo the constant identity-fibre factors. This is a literal line-bundle pullback identity, and specialising $z = 0$ and cancelling the constant factors gives the theorem of the square on the first two factors, proving (a) over $K$ without any Picard or duality input.

2.1F1F3step 1.1construct

Now let $L$ be an invertible sheaf on $H$ and consider the square defect line bundle on $H^{0} \times_{R} H^{0} \times_{R} H$ : the restriction of the square identity to the generic fibre is supplied by step 1.1 for the abelian generic fibre, and the difference of the two sides extends to a line bundle on the smooth total space. Extend the generic base line bundle on $H^{0} \times_{R} H^{0}$ by regular divisor closure using [F3]: the closure of a generic Cartier divisor is Cartier because the regular local rings of the smooth total spaces are UFDs, and Hartogs extends the defining equations in codimension one. The residual square obstruction is then a line bundle with a vertical divisor.

3.1F2F3step 2.1algebra

Because $H^{0}$ has geometrically irreducible fibres by [F2], every vertical prime divisor on $H^{0} \times_{R} H^{0} \times_{R} H$ is the inverse image of a special-fibre component of $H$ , hence is pulled back from the last factor. Restricting the square obstruction to $(unit, unit, {id}_{H})$ makes the square identity trivial, so the residual line bundle pulled back from $H$ is pulled back from $R$ ; it can therefore be absorbed into the base line bundle on $H^{0} \times_{R} H^{0}$ . Hence the square identity holds for $L$ on $H$ with the $H^{0}$ -translation action, proving (b).

4.1F1F2step 3.1algebra∎

The argument uses only the published cube theorem, divisor closure in regular total spaces and the component structure of [F2]; no Picard scheme, dual abelian variety, Chevalley decomposition or Raynaud theorem is used. The same statement applies after the base changes used later, since the hypotheses are stable under flat base change of DVRs.

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Hilbert divisor charts and the Picard diagonal

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be a projective geometrically integral scheme over a field $k$ . Write $P_{A / k}$ for the fppf sheafification of $T \mapsto Pic (A_{T}) / p_{T}^{*} Pic (T)$ . When a rational point $e \in A (k)$ is supplied, normalization along $e$ identifies it with the sheaf of $e$ -rigidified line bundles; for an abelian variety this is The rigidified relative Picard functor and the dual abelian variety. Then:

(a) sufficiently positive relative effective Cartier divisors with fixed Hilbert polynomial form a finite-type open Hilbert chart $D$ ;

(b) on every test $T$ of the appropriate open positive-class subfunctor, the pullback of $D^{+} \to P_{A / k}$ is a smooth proper surjective $T$ -scheme, fppf-locally a projective-space bundle. If the test class is represented by a line bundle $L$ on $A_{T}$ , the pullback is $P ((p_{T, *} L)^{\lor})$ . In general it can be a nonsplit form of projective space; when a rational point $e$ is supplied, rigidification removes this obstruction;

(c) the diagonal of the Picard sheaf is represented and quasi-compact, and the Picard scheme is separated once representability holds.

Facts & Assumptions

Given: AC and DC, a projective geometrically integral $k$ -scheme $A$ , and a very ample line bundle $H$ on $A$ .

[F1]

The Hilbert scheme represents projective flat families with fixed Hilbert polynomial, and its relative effective-divisor locus is open: on a flat finitely presented family, being cut out fibrewise by a regular element with invertible ideal is open by the local flatness criterion and Nakayama, the bad locus being closed and proper over the base (Projective Hilbert schemes represent all flat finitely presented families, Noetherian fibrewise flatness for a module finite over the target, Assuming the Axiom of Choice, Nakayama's lemma, Proper morphisms are closed, Constructible images for finite-presentation affine maps).

[F2]

Relative Castelnuovo-Mumford regularity conditions are finitely many higher-cohomology vanishings, the universal finite cohomology complex computes them compatibly with base change, and regularity propagates to all required nonnegative twists; Serre vanishing makes every individual test family locally lie in such a chart (Regularity gives generation, multiplication, and vanishing, Universal finite projective cohomology complex over any base, Cohomology and base change for proper flat coherent families, Serre global-generation criterion for ampleness, Ampleness is invariant under positive powers, Absolute ampleness by affine section opens).

[F3]

A flat equivalence relation of finite type with a monomorphism to the square, on a separated finite-type scheme with flat projections, has a saturated open with quotient (A flat finite-type equivalence relation has a generic scheme quotient); arbitrary test families descend to finitely generated algebras by finite-presentation spreading (Finite-stage descent of finitely presented schemes and their morphisms, Faithfully flat descent of modules and affine algebras is effective).

[F4]

Geometrically integral proper fibres have only scalar global functions, and their nonzero sections of invertible sheaves are regular (Global functions on proper integral schemes form a finite extension of the base field). Over arbitrary test algebras, a finite affine Cech cover of the separated $k$ -scheme $A$ gives $Γ (A_{T}, O) = Γ (T, O)$ , since tensoring over $k$ preserves its equalizer. The nonempty smooth locus of a geometrically integral finite-type $k$ -scheme has a point over a finite separable extension (A nonempty smooth scheme has a finite separable point). Finite free universal cohomology complexes and affine algebra descent represent and descend the isomorphism locus below; the Picard functor is sheafified as above (Fppf sheaves of sets and sheafification, Sheafification exists for the fppf site, projective space points).

Proof

technique · direct: realize divisor classes in Hilbert charts, represent the linear-system fibres by projective spaces, quotient by linear equivalence, and control the diagonal

1.1F1F2givenconstruct

Fix the very ample $H$ on $A$ . The relative effective-divisor functor is an open subscheme of the Hilbert scheme by [F1]: being cut out by a fibrewise regular element with invertible ideal is open, and for a fixed Hilbert polynomial the divisor scheme $D$ is finite type and quasi-projective. Impose a fixed Castelnuovo-Mumford regularity bound by finitely many higher-cohomology vanishings with respect to $H$ ; by [F2] these vanishings propagate to all nonnegative twists and are computed by the universal finite cohomology complex, so they cut out an open positive chart $D^{+}$ ; do not define openness by an infinite intersection of vanishings. Serre vanishing ensures that after twisting by a sufficiently high power of $H$ every individual test family lies locally on its base in such a chart. A line bundle satisfying these conditions has finite locally free sections of positive rank, compatibly with every base change.

2.1F2F3F4step 1.1construct

Given $c \in P_{A / k} (T)$ , choose an fppf covering $T^{'} \to T$ on which $c$ has a line-bundle representative $L$ . On $T^{'}$ its divisor fibre is $P ((p_{*} L)^{\lor})$ : fibrewise nonzero sections, including twists by base line bundles, give exactly the relative effective Cartier divisors, since the geometric fibres are integral. Positivity makes $p_{*} L$ locally free of positive rank and compatible with base change by step 1.1. The two pullbacks of these projective bundles have canonical identifications, because both represent the same divisor-class fibre functor; they satisfy the cocycle condition. These projective spaces descend to a scheme: locally their common rank is $r$ , the canonical relative anticanonical bundle is $O (r)$ and its section algebra and homogeneous embedding equations descend by faithfully flat module/algebra descent. Taking the descended relative Proj gives a projective scheme whose pullback is the projective bundle; the canonical identifications glue these schemes on $T$ . Its smoothness, properness and surjectivity follow from the projective-space description after the cover (flatness descends and the geometric fibres are projective spaces). A representative on $T$ itself gives the displayed projective bundle directly. If $e$ exists, normalize representatives along $e$ ; scalar global functions make every rigidified isomorphism unique, so their descent data satisfy the cocycle condition and descend a line bundle on $A_{T}$ . Without $e$ this last conclusion is not asserted. This proves (b) on arbitrary tests.

3.1F3step 2.1algebra

The linear-equivalence relation $R \subseteq D^{+} \times_{k} D^{+}$ is represented: two divisor families are equivalent when their classes agree, whose projections are smooth projective bundles: the test $D^{+}$ carries the universal divisor line bundle, so the represented-case clause of step 2.1 applies. It is a monomorphism into the square. Hence the generic quotient theorem [F3] applies and yields a nonempty saturated open $W \subseteq D^{+}$ with a quotient representing the corresponding open subfunctor of the Picard sheaf; all-test statements reduce to finitely generated test algebras by finite-presentation spreading.

4.1F2F3F4step 3.1algebra

First suppose a point $e \in A (k)$ is supplied. On a Noetherian product $T$ of divisor charts normalize $M = L_{1}^{- 1} \otimes L_{2}$ along $e$ . Apply [F2] to $M$ and $M^{- 1}$ , locally choosing finite free complexes in nonnegative degrees. Their degree-zero kernels represent sections on every test, and evaluation at $e$ is represented by linear chain maps to $O_{T}$ (a finite free complex permits such a representative). Impose the finitely many linear equations $d (s) = 0$ , $d (t) = 0$ , $e (s) = e (t) = 1$ in the product of the two degree-zero affine vector bundles. The product $s t$ is a global function, hence a scalar by [F4], and its value at $e$ is $1$ , so $s t = 1$ . Thus this closed affine scheme represents exactly the unique rigidified isomorphism $L_{1} \to L_{2}$ when one exists. It represents the diagonal and is of finite presentation. For general $A$ , pass to a finite separable extension having a rational point by [F4]. The affine representations of the equality-of-classes functor carry canonical descent data, even when the chosen rational points differ on overlaps; affine descent [F3] makes the diagonal affine over $T$ . Finite-presentation spreading and fppf-local representatives extend this conclusion from chart products to arbitrary tests.

5.1F2F3F4step 4.1algebra

To check separatedness once representability holds, apply the valuative criterion. After a faithfully flat field extension a rational section is available, so normalize a line bundle $M$ on $A_{V}$ for a DVR $V$ whose class is generically zero. Choose inverse generic trivializing sections. Proper coherent cohomology makes their section modules finite over $V$ , and flatness of the line bundles injects them into the generic section spaces. Rescale each generic section by a power of the uniformizer until it extends and its reduction is nonzero: a finite torsion-free module over a DVR is a lattice, and the exact sequence for multiplication by the uniformizer makes the reduction map on global sections injective. On the integral special fibre the product of these two nonzero sections is nonzero. Their global product is a scalar by [F4], so it is a unit of $V$ . The extended sections are therefore inverse trivializations after rescaling by that unit; normalization at the section makes the generic isomorphism extend uniquely. This proves the valuative criterion, and hence separatedness of the locally finite-type Picard representative.

6.1F1F2F3step 4.1algebra∎

Finally the universal Hilbert ideal is flat over its Noetherian chart base, and on a fibre where it is a line bundle the Noetherian fibrewise-flatness criterion makes it flat and the finite-flat local-freeness criterion makes it rank one; the locus is open, and proper projection of its failure gives the divisor open in the Hilbert chart. Arbitrary test families descend locally to finitely generated $k$ -algebras by finite-presentation spreading of the line bundle and its inverse, so the representing opens and linear-system universal properties apply to every test.

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Multiplication by n on an abelian scheme is finite flat, and etale for n invertible

Statement

Assume AC and DC as inherited from the finite-flatness and flatness-by-fibres suppliers. Let $S$ be locally Noetherian, let $A \to S$ be an abelian scheme of relative dimension $g$ (Abelian schemes over a base), let $n \geq 1$ , and let $[n] : A \to A$ be multiplication by $n$ . Then $[n]$ is finite, flat and surjective of degree $n^{2 g}$ , and $A [n] = \ker [n]$ is a finite flat $S$ -group scheme of rank $n^{2 g}$ . If $n$ is invertible on $S$ , then $[n]$ is etale and $A [n] \to S$ is finite etale of rank $n^{2 g}$ .

Facts & Assumptions

Given: AC and DC, a locally Noetherian base $S$ , an abelian scheme $A \to S$ of relative dimension $g$ , and $n \geq 1$ .

[F1]

On every geometric fibre, multiplication by $n$ is finite, flat and surjective with kernel of order $n^{2 g}$ ; on an abelian variety, $[n]$ is a finite faithfully flat isogeny of degree $n^{2 g}$ (Nonzero multiplication on an abelian variety is finite and faithfully flat).

[F2]

Properness and quasi-finiteness imply finiteness; quasi-finiteness is checked on fibres (A proper quasi-finite morphism is finite, Finite-fibre and pointwise characterizations of quasi-finiteness); flatness of a finite morphism can be checked fibrewise in the Noetherian setting (Noetherian fibrewise flatness for a module finite over the target); etaleness of an equal-relative-dimension morphism is detected by invertibility of the differential determinant (Étale equals flat and unramified in finite presentation, Differentials of a smooth morphism, Étale morphism of schemes).

[F3]

The group law of $A$ is commutative, so $[n]$ is a group homomorphism, and fibrewise structures are as in Fibres of abelian schemes and unit-preserving morphisms; base change and products preserve abelian schemes (Base change and products of abelian schemes).

Proof

technique · direct: fibrewise finiteness, then flatness by the fibrewise criterion, then etaleness from the differential

1.1F1F2givenalgebra

By [F1] every geometric fibre of $[n] : A \to A$ has finite kernel of order $n^{2 g}$ , so $[n]$ is quasi-finite; it is proper because $A$ is proper over $S$ , hence finite by [F2]. Consequently $A [n] = \ker [n]$ is finite over $S$ and of finite type.

2.1F1F2step 1.1algebra

Flatness of $[n]$ follows by applying the Noetherian flatness-by-fibres criterion [F2] to the local tower $O_{S, s} \to O_{A, y} \to O_{A, x}$ for $[n]$ , with $M = O_{A, x}$ : smoothness makes $M$ flat over $O_{S, s}$ , and the field-level multiplication theorem makes the special-fibre module flat over the special-fibre target. Constancy of the rank then follows from the rank $n^{2 g}$ on geometric fibres, so $[n]$ is finite flat of degree $n^{2 g}$ and $A [n]$ has rank $n^{2 g}$ .

3.1F2F3step 2.1algebra∎

If $n$ is invertible on $S$ , then the differential of $[n]$ at the identity is multiplication by the unit $n$ on the locally free sheaf of invariant differentials (the differential of the group law is addition), and translation-equivariance spreads this to every point; the equal-relative-dimension criterion of [F2] therefore makes $[n]$ etale, and $A [n] \to S$ , being the pullback along the identity section, is finite etale of rank $n^{2 g}$ .

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An abelian scheme is the Neron model of its generic fibre

Statement

Assume AC and DC. Let $S$ be a Dedekind scheme with function field $K$ , and let $A \to S$ be an abelian scheme (Abelian schemes over a base). Then $A$ is a Neron model (Neron models, the Neron mapping property and weak Neron models) of its generic fibre $A_{K}$ : for every smooth $S$ -scheme $Y$ and every $K$ -morphism $u_{K} : Y_{K} \to A_{K}$ there is a unique $S$ -morphism $Y \to A$ extending $u_{K}$ .

Facts & Assumptions

Given: AC and DC, a Dedekind scheme $S$ with function field $K$ , an abelian scheme $A \to S$ , a smooth $S$ -scheme $Y$ , and a $K$ -morphism $u_{K} : Y_{K} \to A_{K}$ .

[F1]

For a smooth finite-type $S$ -scheme $Z$ , every $K$ -morphism $Z_{K} \to A_{K}$ extends uniquely to $Z \to A$ (K-morphisms from smooth models into abelian schemes extend uniquely).

[F2]

A smooth morphism is locally of finite presentation; over the locally Noetherian scheme $S$ , every point of a smooth $S$ -scheme has an open neighbourhood of finite type over $S$ (Smooth morphism of schemes, Locally Noetherian and Noetherian schemes).

[F3]

Two $S$ -morphisms from a flat $S$ -scheme to a separated $S$ -scheme agreeing on the generic fibre are equal. Indeed their equalizer is closed; on a chart over an affine integral open $Spec B \subseteq S$ , its ideal vanishes after tensoring with $K$ . Flatness makes the chart ring $B$ -torsion-free, so that ideal is zero. This argument uses the closed diagonal (Separated morphism of schemes) and generic localization (Scheme-theoretic fibre); it does not require the generic fibre to be open.

Proof

technique · apply the extension result for finite-type smooth tests, then cover an arbitrary smooth test by finite-type opens and glue using separatedness

1.1F1givenconstruct

First suppose $Y$ is of finite type over $S$ . Then [F1] gives the unique extension $u : Y \to A$ of $u_{K}$ . This proves the mapping property for finite-type smooth test schemes.

2.1F1F2F3step 1.1construct∎

For an arbitrary smooth $S$ -scheme $Y$ , use [F2] to cover it by open subschemes $Y_{i}$ of finite type over $S$ . Apply step 1.1 to each restriction $u_{K} ∣_{(Y_{i})_{K}}$ , obtaining $u_{i} : Y_{i} \to A$ . On an overlap $Y_{i} \cap Y_{j}$ , the maps agree on the schematically dense generic fibre, so they agree everywhere by [F3]. The $u_{i}$ glue to an $S$ -morphism $u : Y \to A$ extending $u_{K}$ . The same density and separatedness give uniqueness. Thus $A$ satisfies the full Neron mapping property; the weak property is a consequence, and no group law on a general model is constructed.

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Separated minimal union and translations

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ , let $R^{s h}$ be a strict henselization, and let $A / K$ be an abelian variety. Then there exists a smooth separated finite-type faithfully flat $R$ -model $X$ of $A$ , formed by gluing finitely many minimal representatives of $A$ along $A$ . Moreover, for $R' = O_{Z, η}$ with $Z$ smooth of finite type over $R$ and $η$ a generic point of its special fibre, every translation of $A_{K'}$ by a point of $A (K')$ , where $K' = Frac R'$ , extends to an $R'$ -birational self-map of $X_{R'}$ which is an open immersion on its $R'$ -dense domain of definition. This assertion supplies a rational map; it does not assert extension over the omitted points.

Facts & Assumptions

Given: AC and DC, a DVR $R$ with fraction field $K$ , residue field $k$ , a strict henselization $R^{s h}$ , and an abelian variety $A / K$ .

[F1]

Invariant top forms on smooth models define an order; the normalized-form comparison proves that a birational rational map whose generic isomorphism preserves the chosen invariant form cannot decrease order, and equality makes it an open immersion on its domain. Smooth models over the regular DVR are regular, hence normal. There are finitely many minimal equivalence classes, and their representatives remain minimal after the indicated smooth-DVR base changes, after splitting special-fibre components (Invariant volume and finite minimal classes, Regularity ascends and descends along a flat local homomorphism, Locally standard smooth iff flat with geometrically regular fibres, regular local rings are normal).

[F2]

A finite weak Neron model collection receives every generic rational map from a smooth DVR model with irreducible special fibre; its weak property and the minimal representatives are compatible with the generic smooth-DVR base changes used here (Projective weak models and rational mapping, Invariant volume and finite minimal classes). Since the special fibres of its smooth finite-type members are regular with finitely many disjoint irreducible components, replacing each member by the finitely many opens consisting of its generic fibre together with one special component preserves the weak property (Smooth morphism of schemes).

[F3]

The schematic closure of the generic diagonal is flat over a DVR, and a morphism to a separated target is determined by its restriction to a schematically dense open of a reduced source (Schematic closure and agreement on a dense open, Agreement on a schematically dense open).

[F4]

Compatible identifications along a common open glue schemes; separatedness is equivalent to the diagonal being a closed immersion, and closed immersions are local on the target (Compatible open pieces of ringed or locally ringed spaces glue, Schemes, Separated morphism of schemes, The diagonal morphism, Closed immersions are local on the target).

[F5]

A separated quasi-finite morphism factors locally as an open immersion followed by a finite morphism; a finite birational algebra over a normal domain is the domain itself (Scheme Zariski Main factorization for separated quasi-finite morphisms, regular local rings are normal).

Proof

technique · follow the separated-minimal-model construction and translation argument of BLR 4.3/4, using invariant-volume comparison for the generic translations

1.1F1F3F5givenalgebra

Choose finitely many representatives $X_{1}, \dots, X_{m}$ of all minimal equivalence classes using [F1]. For $i \neq j$ , let $Γ_{i j}$ be the schematic closure of the generic diagonal $A ↪ X_{i} \times_{R} X_{j}$ . It is integral and flat over $R$ by [F3]. Suppose its special-fibre support has dense image in $(X_{i})_{k}$ . The ambient product is regular of dimension $2 g + 1$ , and the generic diagonal has codimension $g$ there; its closure therefore has dimension $g + 1$ . A component of $(Γ_{i j})_{k}$ has dimension at most $g$ , since it is a height-one component cut out by the nonzero divisor $π$ . A component dominating the $g$ -dimensional $(X_{i})_{k}$ is consequently generically finite over it. At its generic point $q$ , the projection $p_{i} : Γ_{i j} \to X_{i}$ is quasi-finite. It is separated and birational, because its generic-fibre map is the identity. On an affine neighbourhood of $ξ = p_{i} (q)$ , [F5] factors it as an open immersion into a finite scheme. The reduced closure of the birational generic component in that finite scheme is finite birational over the normal coordinate ring of $X_{i}$ and lies in its fraction field, so normality makes it equal to that coordinate ring. Thus $p_{i}$ is an isomorphism near $q$ and $ξ$ . The other projection then gives an $R$ -birational map between $X_{i}$ and $X_{j}$ ; their minimality and [F1] imply they are equivalent, contrary to their representing distinct classes. The same argument applies to the projection to $X_{j}$ . Therefore both special-fibre projection images are nowhere dense. Remove their closures from the special fibres, for all finitely many pairs, and write $X_{i}^{\circ}$ for the resulting open models. The generic diagonal is now closed in each $X_{i}^{\circ} \times_{R} X_{j}^{\circ}$ for $i \neq j$ .

2.1F3F4givenstep 1.1construct

Glue the $X_{i}^{\circ}$ along their common open generic fibre $A$ by the identity. The identity and cocycle conditions hold because every overlap is the same $A$ . The resulting scheme $X$ is smooth and of finite type over $R$ , since these properties hold on its open cover by the $X_{i}^{\circ}$ . Its diagonal is a closed immersion: on $X_{i}^{\circ} \times_{R} X_{i}^{\circ}$ this follows from separatedness of $X_{i}^{\circ}$ , and on $X_{i}^{\circ} \times_{R} X_{j}^{\circ}$ for $i \neq j$ its image is the closed generic diagonal established in step 1.1; closedness is local on the target by [F4]. Every special fibre remains nonempty after removing nowhere-dense closed subsets, so $X \to Spec R$ is surjective; it is flat because it is smooth. Thus $X$ is a smooth separated finite-type faithfully flat $R$ -model of $A$ .

3.1F1F2F3F5step 1.1step 2.1algebra

First take $R^{'} = R$ . Let $C$ be an irreducible component of the special fibre of $X$ , and let $U_{C} = A \cup C$ , an open model with irreducible special fibre. By [F1] it is minimal. Split the finite weak model collection in [F2] into open models with irreducible special fibre; this preserves its weak property. For $a \in A (K)$ , apply [F2] to $t_{a} : A \to A$ to obtain an $R$ -rational map $f : U_{C} ⇢ Y$ into one such member, with generic fibre $t_{a}$ . Let $ω$ be the invariant top form on $A$ . On the domain of $f$ , the pullback of the normalized generator $π^{- ord (Y)} ω$ is $π^{ord (U_{C}) - ord (Y)}$ times the normalized generator on $U_{C}$ , because $t_{a}^{*} ω = ω$ . Regularity of this pullback gives $ord (U_{C}) \geq ord (Y)$ . Minimality of $U_{C}$ gives the reverse inequality, so the orders are equal. The pullback of the normalized top form is then a unit, so the relative differential determinant is a unit and $f$ is etale on its domain. It is separated and birational, hence quasi-finite; [F5] and normality of $Y$ show it is an open immersion there. In particular $Y$ is minimal and belongs to one of the finitely many classes represented in step 1.1. Composing with the representative's identity birational map gives an open-immersion extension of $t_{a}$ into $X$ on that domain. Doing this for each $C$ gives compatible rational maps because their generic restrictions are all $t_{a}$ ; they glue to an $R$ -rational self-map of $X$ . On its domain the glued map still pulls back the normalized invariant top form to a unit, so it is etale; it is birational, and [F5] makes it an open immersion. The same construction for $t_{- a}$ gives the inverse birational map, since the composites agree with the identity on $A$ and [F3] gives dense-open agreement.

4.1F1F2step 3.1algebra∎

For a generic smooth-DVR extension $R^{'} = O_{Z, η}$ , where $Z$ is smooth and of finite type over $R$ and $η$ is a generic point of its special fibre, base change the construction to $R^{'}$ . The weak model property and the representatives' minimality persist by [F1, F2], after splitting the special fibres into their irreducible components. The argument of step 3.1 therefore applies to each component of $X_{R^{'}}$ and every $a \in A (K^{'})$ , where $K^{'} = Frac (R^{'})$ , giving the claimed $R^{'}$ -birational open immersion.

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Divisor ampleness and quasi-projectivity of group models

Statement

(a) Every effective Cartier divisor $D$ on $H$ whose complement is affine and fibre-dense gives an ample invertible sheaf $O (D)$ .

(b) $H$ is quasi-projective over $R$ ; the embedding is produced by an explicit affine-section chart construction and does not apply a proper-source very-ample theorem to $H$ .

Facts & Assumptions

Given: AC and DC, a DVR $R$ , a smooth separated finite-type $R$ -group scheme $H$ with abelian generic fibre, and an effective Cartier divisor $D$ with affine fibre-dense complement.

[F1]

A finite cover by affine nonvanishing loci of positive-power sections makes a line bundle ample (Absolute ampleness by affine section opens). For the faithfully flat base extension $R \to R^{s h}$ , ampleness descends as follows. For any coherent $F$ on the separated finite-type $R$ -scheme, global sections commute with this flat base change: a finite affine cover and its affine intersections compute sections by a finite equalizer. If the pulled-back line bundle is ample, the pulled-back twists of $F$ are globally generated for all sufficiently large powers by Serre global-generation criterion for ampleness. The evaluation map downstairs pulls back to that surjective evaluation map; its cokernel is zero by Descent of vanishing along a faithfully flat morphism. Thus all these twists are globally generated downstairs, and Serre's criterion gives ampleness.

[F2]

Sections over the nonvanishing locus of a section extend after multiplying by powers of that section, and affine-locus covers produce projective embeddings; closed-immersion locality on the target and stable positive powers are available (Extend a quasi-coherent section after multiplying by a power, Generating line-bundle sections define a morphism to projective space, Closed immersions are local on the target, Ampleness is invariant under positive powers, Immersion of schemes).

[F3]

The identity component $H^{0}$ has geometrically connected fibres with orbits the connected components, the theorem of the square holds on $H$ for the $H^{0}$ -action, and the affine codimension-one neighbourhood lemma supplies $R$ -dense affine opens with effective horizontal Cartier boundary (Cube-derived square over DVR, Affine codimension-one neighbourhoods and divisors, Strict henselization of a DVR and smooth sections).

Proof

technique · direct: fibre-density lets the square produce enough sections of $\mathcal O(D)$ to cover $H$, and the affine-section chart construction embeds the model projectively

1.1F1F3givenalgebra

By [F1] it suffices for ampleness to exhibit a finite cover of $H$ by affine nonvanishing loci of sections of positive powers of $O (D)$ , or to descend the same statement along a faithfully flat extension. Fibre-density of the complement means that $H ∖ D$ meets every $H^{0}$ -orbit, and after base change to a strict henselization the theorem of the square [F3] gives a linear equivalence $D_{g} + D_{g^{- 1}} \sim 2 D$ for suitable translates. The associated sections have nonvanishing loci $g U \cap g^{- 1} U$ , where $U = H ∖ D$ ; for a prescribed geometric point, the two conditions on $g$ define dense opens in the geometrically integral $H^{0}$ -fibre. Section values are dense there by [F3], including after extension of its field: an evaluation injection into the product of the fields of section values stays injective after field extension, as can be checked using finitely many linearly independent coefficients. Hence some section $g$ meets both conditions. These loci cover $H$ , and quasi-compactness extracts a finite subcover. Each such locus is affine: it is the intersection of two affine opens in the separated scheme over the affine base. The affine-locus definition in [F1] gives ampleness of $O (D)$ over the strict henselization, and fpqc ampleness descent in [F1] gives it over $R$ . This proves (a).

2.1F1F2step 1.1construct

For (b), choose an $R$ -dense affine open supplied by the codimension-one neighbourhood lemma and its effective horizontal Cartier boundary $D$ ; by (a) $O (D)$ is ample. Choose finitely many affine section opens $X_{s_{i}}$ covering $H$ and raise the $s_{i}$ to a common positive degree; each $Γ (X_{s_{i}}, O)$ is a finite-type $R$ -algebra with finitely many generators $f_{i j}$ , and the section-extension lemma [F2] extends each $f_{i j} s_{i}^{n}$ to a global section of a sufficiently large common power of $O (D)$ , whose ratios to $s_{i}^{n}$ are exactly $f_{i j}$ on $X_{s_{i}}$ .

3.1F2step 2.1algebra∎

Including the $s_{i}^{n}$ in a finite global section list with no common zero defines a map $H \to P_{R}^{M}$ by [F2]; on the projective chart for $s_{i}^{n}$ its preimage is $X_{s_{i}}$ and the coordinate-ring map is surjective because the ratios include all generators $f_{i j}$ , so the map is a closed immersion into the union of these projective charts by closed-immersion locality. That union is open in projective space, so the map is a locally closed immersion and $H$ is quasi-projective over $R$ ; no properness of $H$ is used. This proves (b).

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Picard representation by generic quotient and translates

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be an abelian variety over an algebraically closed field $k$ . Then the rigidified relative Picard functor of $A$ (The rigidified relative Picard functor and the dual abelian variety) is represented by a separated locally finite-type $k$ -group scheme, with a universal rigidified invertible sheaf, on every test scheme, including nonreduced tests.

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ over an algebraically closed field $k$ , and the rigidified Picard sheaf $P$ of $A$ .

[F1]

The rigidified functor is an fppf sheaf, normalization identifies classes with $Pic (A_{T}) / p_{T}^{*} Pic (T)$ , and rigidified bundles have no nontrivial automorphisms (Rigidification and effective descent of line bundles, assuming AC and DC).

[F2]

The Hilbert divisor charts give an open positive chart $D^{+}$ mapping to $P$ as a relatively projective-space bundle, with a represented flat finite-type linear-equivalence relation and a saturated quotientable open $W$ whose quotient $Y$ is an open subfunctor of $P$ (Hilbert divisor charts and the Picard diagonal, A flat finite-type equivalence relation has a generic scheme quotient).

[F3]

Represented fppf sheaves glue along open subfunctors, and closed subsets of finite-type $k$ -schemes are detected on closed points with residue field $k$ by the Nullstellensatz (Scheme morphisms satisfy fppf descent, Gluing affine schemes along compatible open isomorphisms, Over an algebraically closed field, every maximal ideal is an evaluation ideal).

Proof

technique · direct: open subfunctor from the generic quotient, translations cover all classes, and the open pieces glue to a global representation

1.1F1F2givenalgebra

Let $V$ be the image of the saturated open $W$ of the divisor chart in $P$ ; by [F2] $V$ is an open subfunctor, not merely a set of geometric classes. For a test $T \to P$ , restrict to the open $T^{+}$ on which the pullback has the fixed Hilbert polynomial and the finite positive-regularity conditions of the chart; polynomial local constancy and the finite-cohomology vanishing conditions make $T^{+}$ open, and finite-presentation descent handles arbitrary tests. Over $T^{+}$ the divisor map is the faithfully flat open projective-space bundle of sections of [F2], and saturation makes the preimage of $W$ invariant under its kernel pair, so it descends to an open of $T^{+}$ and hence of $T$ ; on that open the pullback is represented by the quotient $Y \times_{P} T$ . The fppf sheaf quotient equality identifies $V$ with $Y$ , giving an open immersion of represented functors $V ↪ P$ .

2.1F2F3step 1.1algebra

Translate $V$ by every rigidified class over $k$ , i.e. consider the subfunctors $V \cdot x$ for $x \in P (k)$ ; every $k$ -valued class lies in such a translate because choosing $v \in V (k)$ (nonempty since $V$ is a nonempty locally finite-type open) gives $x = (x - v) + v$ . For an arbitrary test, pull the union of the translates back to each finite-type positive divisor chart of [F2] for every polynomial and sufficiently high twist: this pullback is open and contains every closed point of the chart, since closed points have residue field $k$ ; its closed complement is therefore empty by the Nullstellensatz [F3]. The positive divisor charts, with twists reversed, cover $P$ fppf-locally on every test: locally a sufficiently positive twist has locally free nonzero sections, and a fibrewise nonzero section exists after the projective-space cover of [F2]. Hence every test pulls back to the union of the translates.

3.1F2F3step 2.1algebra∎

The represented open overlaps of the translates with identity transition maps glue along the open cover of step 2.1 to a scheme representing the full functor $P$ on all tests, including nonreduced and non-Noetherian tests, by the gluing and descent statements of [F3]; the universal rigidified invertible sheaf is obtained by gluing the universal bundles of the chart quotients, and separatedness was proved on the divisor charts in [F2]. This chart argument does not assume that every geometric class descends to $k$ .

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Good reduction supplies a Neron model

Statement

Assume AC and DC. Let $S$ be a Dedekind scheme with function field $K$ and let $A_{K}$ be an abelian variety over $K$ with good reduction over $S$ (Good reduction of an abelian variety over a Dedekind scheme). Then $A_{K}$ admits a Neron model over $S$ , namely any abelian scheme model $A \to S$ of $A_{K}$ , and this model is unique up to a unique $S$ -isomorphism inducing the specified identity on $A_{K}$ . In particular, for a discrete valuation ring $R$ with fraction field $K$ , every abelian variety over $K$ with good reduction has a Neron model over $R$ , and that Neron model is proper and smooth over $R$ .

Facts & Assumptions

Given: AC and DC, a Dedekind scheme $S$ with function field $K$ , an abelian variety $A_{K} / K$ with good reduction, and an abelian scheme model $A \to S$ of $A_{K}$ .

[F1]

By definition of good reduction there is an abelian scheme $A \to S$ with $A \times_{S} Spec K ≅ A_{K}$ (Good reduction of an abelian variety over a Dedekind scheme).

[F2]

An abelian scheme over a Dedekind scheme is a Neron model of its generic fibre, and Neron models are unique up to a unique isomorphism over the generic fibre (An abelian scheme is the Neron model of its generic fibre, Neron models, the Neron mapping property and weak Neron models).

Proof

technique · direct

1.1F1F2givenalgebra

Let $A \to S$ be an abelian scheme model of $A_{K}$ , supplied by [F1]. By [F2] $A$ satisfies the Neron mapping property; since $A \to S$ is smooth, separated and of finite type, it is a Neron model of $A_{K}$ .

2.1F2step 1.1algebra∎

Any two Neron models of $A_{K}$ are related by a unique $S$ -isomorphism inducing the specified identity on $A_{K}$ , by the uniqueness clause of [F2], so the model is unique up to unique isomorphism over the specified generic fibre; it is proper and smooth because it is an abelian scheme. In the DVR case the same statement applies to $S = Spec R$ .

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Good reduction is stable under base change of the base

Statement

Assume AC and DC. Let $S$ be a Dedekind scheme with function field $K$ , let $A_{K}$ be an abelian variety with good reduction over $S$ witnessed by an abelian scheme $A \to S$ , and let $S^{'} \to S$ be a dominant morphism of Dedekind schemes with function field $K^{'}$ (for instance the normalization of $S$ in a finite extension $K^{'} / K$ , or the localisation of $S$ at a point). Then $A_{S^{'}} \to S^{'}$ is an abelian scheme and $A_{K^{'}} = A_{S^{'}} \times_{S^{'}} Spec K^{'}$ has good reduction over $S^{'}$ ; the model is the base change of the model $A$ . In particular good reduction is preserved by finite extensions of the function field and by localisation of the base. Nothing is asserted about the converse: descent along ramified base change can fail, as recorded on the companion examples page.

In addition, every abelian variety over a number field $K$ has good reduction at all but finitely many finite places.

Facts & Assumptions

Given: AC and DC, a Dedekind scheme $S$ with function field $K$ , an abelian scheme $A \to S$ , a dominant morphism $S^{'} \to S$ of Dedekind schemes, and, for the second clause, an abelian variety over a number field $K$ .

[F1]

Abelian schemes are stable under base change: $A_{S^{'}} \to S^{'}$ is an abelian scheme of the same relative dimension, with generic fibre $A_{K^{'}}$ (Base change and products of abelian schemes, Base change of objects, morphisms and properties, Good reduction of an abelian variety over a Dedekind scheme).

[F2]

Objects and morphisms of finite presentation descend along filtered colimits, and properness descends along such stages (Finite-stage descent of finitely presented schemes and their morphisms, Finite-stage descent of properness for finitely presented schemes); the ring of integers is a free $Z$ -module of rank the degree and a Dedekind domain, and algebraic numbers have bounded denominators (The ring of integers has rank the degree, Rings of integers are Dedekind domains, Clearing denominators for an algebraic number).

[F3]

The smooth locus is open, and the perfect complex of cohomology of a proper flat finitely presented sheaf is compatible with base change; a proper geometrically integral fibre has global functions equal to its base field (The smooth locus is open, Proper morphisms are closed, Universal finite projective cohomology complex over any base, Global functions on proper integral schemes form a finite extension of the base field).

Proof

technique · direct for base change; spreading with finitely many denominator conditions for the number-field clause

1.1F1givenalgebra

By [F1] the base change $A_{S^{'}} = A \times_{S} S^{'} \to S^{'}$ is an abelian scheme and its generic fibre is $A_{K^{'}}$ ; hence $A_{K^{'}}$ has good reduction over $S^{'}$ with model $A_{S^{'}}$ , and since $S^{'} \to S$ is dominant the function field extension is defined. This proves the base-change stability statements, and no converse assertion is made.

2.1F2F3step 1.1construct

For the number-field clause write $K$ as the filtered colimit of the rings $O_{K} [1 / d]$ , $d \neq 0$ ; by [F2] the finitely presented abelian variety $A_{K}$ descends to a proper finitely presented model over some $O_{K} [1 / d]$ , and multiplication, unit, inverse and their finitely many identities descend at a common later stage. The smooth locus of the descended model is open, and its closed nonsmooth locus has closed image under the proper structure morphism and misses the generic point, so inverting one further integer removes it. Smoothness over the Dedekind base then gives flatness of the structure sheaf, and the removed closed set is finite.

3.1F2F3step 2.1algebra

Choose the universal finite projective cohomology complex of the structure sheaf over a stage by [F3]. Over $K$ its differentials split, so the complex is the direct sum of its finite-dimensional cohomology and contractible pairs; all bases, inverse matrices and chain identities involve finitely many denominators and spread after inverting one further integer. The degree-zero remaining module has rank one because $H^{0} (A_{K}, O) = K$ by [F3]. Therefore every geometric fibre has degree-zero cohomology of dimension one by universal cohomology comparison and is connected: a disconnected proper fibre would produce independent nontrivial clopen idempotents. Smoothness gives relative dimension $g$ after discarding any components absent generically, whose closed proper images miss the generic point.

4.1F1F2step 3.1algebra∎

The resulting smooth proper finitely presented group scheme with geometrically connected fibres is an abelian scheme by Abelian schemes over a base; the places removed form a finite set of closed points of $Spec O_{K}$ , and localizing at every remaining place proves that $A_{K}$ has good reduction there. This supplies the number-field spreading clause directly rather than using base-change stability as a substitute for spreading.

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Connected smooth quasiprojective model with proper special fibre is proper

Statement

Assume AC and DC as inherited from the stated suppliers. Let $R$ be a discrete valuation ring and let $G$ be a smooth separated finite-type quasi-projective $R$ -scheme whose generic fibre is an abelian variety. If the special fibre $G_{k}$ is proper and geometrically connected, then $G$ is proper over $R$ . Applied to the connected identity model of a smooth group scheme with abelian generic fibre, $G$ becomes an abelian scheme.

Facts & Assumptions

Given: AC and DC, a DVR $R$ with residue field $k$ , uniformizer $t$ , and a smooth separated finite-type quasi-projective $R$ -scheme $G$ with abelian generic fibre and proper geometrically connected special fibre.

[F1]

The quasi-projective identity model with abelian generic fibre is Divisor ampleness and quasi-projectivity of group models; schematic closures are contracted from the generic fibre, and a smooth scheme is flat hence schematically dense (Schematic closure and agreement on a dense open, Over a principal ideal domain flatness is equivalent to torsion-freeness, Every DVR is a PID).

[F2]

The perfect-complex machinery for a proper flat finitely presented morphism: the cohomology of $O$ is computed by a bounded finite projective complex concentrated in degrees $\geq 0$ , with naturality in base algebras (Universal finite projective cohomology complex over any base); Noetherian completion is flat and faithfully flat, and properness descends along faithfully flat maps (The completion of a Noetherian ring is flat, Completion of a finite module is extension of scalars, A flat ring map is faithfully flat exactly when it detects proper ideals and is surjective on spectra, Properness descends through fpqc base change); a morphism from a proper source to a separated target is proper (Morphisms from a proper scheme to a separated one are proper), and proper integral fibres have constant functions (Global functions on proper integral schemes form a finite extension of the base field, Projective coherent finiteness and large twist vanishing).

Proof

technique · direct: prove properness over the completion by a connectedness/idempotent argument on the projective closure, then descend to $R$

1.1F1F2givenalgebra

Properness descends along the faithfully flat completion $R \to \hat{R}$ by [F2], so it suffices to treat a complete DVR. In that case view $G$ as an open subscheme of its projective schematic closure $X$ in $P_{R}^{m}$ by [F1]; the closure is $t$ -torsion-free on coordinate charts because its ideal is contracted from the generic fibre, so $X$ is flat over $R$ , and $X_{K} = G_{K}$ because the generic fibre $G_{K}$ is proper, hence closed in projective space. The finite $R$ -module $H^{0} (X, O_{X})$ injects into $H^{0} (X_{K}, O) = K$ by flatness, and every element is integral over the integrally closed ring $R$ , so $H^{0} (X, O_{X}) = R$ .

2.1F2step 1.1algebra

If $X_{k}$ were disconnected, a nontrivial clopen partition would define compatible idempotents $e_{n} \in H^{0} (X_{n}, O)$ on the nilpotent thickenings $X_{n} = X \times_{R} R / t^{n + 1}$ , which share the same underlying space. Applying the finite projective complex of [F2] to the proper flat $X$ and $O_{X}$ gives $H^{0} (X_{n}, O) = \ker (C_{0} / t^{n + 1} \to C_{1} / t^{n + 1})$ ; finite projective modules over complete $R$ are complete and inverse limits preserve kernels, so $\underset{\leftarrow}{l i m} H^{0} (X_{n}, O) = \ker (C_{0} \to C_{1}) = H^{0} (X, O) = R$ . The compatible idempotents would therefore produce a nontrivial idempotent in $R$ , impossible; hence $X_{k}$ is connected.

3.1F2step 2.1algebra

The open immersion $G_{k} \to X_{k}$ has proper source and separated target, hence is proper by [F2], so its image is closed and open and nonempty in the connected $X_{k}$ ; it is therefore all of $X_{k}$ , and $G_{k} \to X_{k}$ is an isomorphism. Since the generic fibres already coincide, the closed complement $X ∖ G$ has empty fibres over both points of $Spec R$ , hence is empty and $G = X$ is proper over $R$ .

4.1F1F2step 3.1algebra∎

Applied to the identity component $G = H^{0}$ of a smooth separated finite-type $R$ -group scheme $H$ with abelian generic fibre: $H^{0}$ is quasi-projective (Divisor ampleness and quasi-projectivity of group models), smooth separated finite type with geometrically connected fibres, so if its special fibre is proper then $G$ is proper by steps 1.1 and 2.1 and is an abelian scheme over $R$ . This lemma does not claim that arbitrary smooth connected group models are quasi-projective, and the argument retains the AC and DC assumptions of the perfect-complex supplier.

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Abelian scheme torsion specialization is unramified

Statement

Assume AC and DC as inherited from the stated suppliers. Let $R$ be a discrete valuation ring with fraction field $K$ , residue field $k$ and strict henselization $R^{s h}$ , let $A \to Spec R$ be an abelian scheme of relative dimension $g$ , and let $ℓ \neq char k$ be prime. Then for every $ν \geq 1$ :

(a) $A [ℓ^{ν}]$ is finite etale over $R$ of rank $ℓ^{2 g ν}$ ;

(b) over $R^{s h}$ , specialization identifies the geometric generic points of $A [ℓ^{ν}]$ with the special separable points, and the inertia group acts trivially on $A [ℓ^{ν}]$ , hence on the Tate module.

Facts & Assumptions

Given: AC and DC, a DVR $R$ with fraction field $K$ , residue field $k$ and strict henselization $R^{s h}$ , an abelian scheme $A / R$ of relative dimension $g$ , and a prime $ℓ \neq char k$ .

[F1]

Multiplication by $ℓ^{ν}$ is finite flat of degree $ℓ^{2 g ν}$ on an abelian scheme, and etale when $ℓ^{ν}$ is invertible on the base; properness and quasi-finiteness imply finiteness (Multiplication by n on an abelian scheme is finite flat, and etale for n invertible, A proper quasi-finite morphism is finite, Prime to characteristic multiplication is etale, Abelian schemes over a base).

[F2]

The geometric torsion of the generic fibre is $(Z / ℓ^{ν})^{2 g}$ and the rank is locally constant (Field prime to characteristic torsion and Tate module).

[F3]

Over a strictly henselian local ring with separably closed residue field, a finite etale scheme splits as a disjoint union of copies of the base, reduction is a bijection on sections, (Strict henselian etale sections). The chosen valuation determines $K^{s h} \subseteq K^{s e p}$ and inertia $I = Gal (K^{s e p} / K^{s h})$ (Prime-to-residue-characteristic Tate modules and inertia, Strict henselization of a DVR and smooth sections).

Proof

technique · direct: finiteness and etaleness of the torsion, then splitting over the strict henselization

1.1F1F2givenalgebra

By [F1] the morphism $[ℓ^{ν}] : A \to A$ is finite, flat and of degree $ℓ^{2 g ν}$ , and since $ℓ$ is invertible on $R$ it is etale; its kernel $A [ℓ^{ν}]$ , the pullback along the identity section, is finite etale of rank $ℓ^{2 g ν}$ by [F2] (the rank is locally constant and equals $ℓ^{2 g ν}$ on the generic fibre of the connected base $Spec R$ ). This proves (a).

2.1F1F3step 1.1algebra∎

Base change to $R^{s h}$ : the scheme $A [ℓ^{ν}]_{R^{s h}}$ is finite etale over the strictly henselian local ring $R^{s h}$ with separably closed residue field $k^{s}$ , so by [F3] it is a disjoint union of copies of $Spec R^{s h}$ ; in particular every geometric generic point is already rational over $K^{s h}$ , and reduction is a bijection between the generic geometric points and the special separable points. Consequently $Gal (K^{s e p} / K^{s h})$ fixes every point of $A [ℓ^{ν}] (K^{s e p})$ , so the inertia group $I = Gal (K^{s e p} / K^{s h})$ acts trivially; the same holds on the inverse limit $T_{ℓ} (A)$ , because inertia acts coordinatewise on the inverse limit and fixes every torsion coordinate. This is the precise finite-etale smooth-proper specialization statement used here; it is not general smooth proper base change for higher cohomology.

Remarks

For a nonhenselian DVR this proves triviality of the chosen inertia subgroup. Equivariance with the residue Galois group concerns the decomposition subgroup of the chosen valuation, not the full absolute Galois group of $K$ .

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Birational group law

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ , let $R^{s h}$ be a strict henselization, let $A / K$ be an abelian variety, and let $X$ be the separated minimal model of Separated minimal union and translations. Then the generic multiplication of $A$ extends to an $R$ -birational associative law $m$ on $X$ whose universal left and right translations are birational. Here an $R$ -birational group law is an $R$ -rational multiplication on $X \times_{R} X$ which is associative wherever the compositions are defined and whose universal translations $(x, y) \mapsto (x, m (x, y))$ and $(x, y) \mapsto (m (x, y), y)$ are $R$ -birational self-maps of $X \times_{R} X$ .

Facts & Assumptions

Given: AC and DC, a DVR $R$ with fraction field $K$ , an abelian variety $A / K$ , the separated minimal model $X$ of $A$ over $R$ , and the generic multiplication $m_{K} : A \times_{K} A \to A$ .

[F1]

Over $R^{'} = O_{Z, η}$ , each translation by an $A (K^{'})$ -point extends to an $R^{'}$ -birational self-map of $X_{R^{'}}$ , an open immersion on its $R^{'}$ -dense domain (Separated minimal union and translations).

[F2]

Domains of $S$ -rational maps, descent of representatives along faithfully flat maps, and equality of morphisms agreeing on a schematically dense open of a reduced source (S-dense open subschemes and S-rational maps, An S-rational map defined after a faithfully flat smooth base change is defined, Scheme morphisms satisfy fppf descent, Agreement on a schematically dense open).

Proof

technique · direct: construct the universal translations at special generic points, spread, and compare

1.1F1F2givenconstruct

Let $ξ$ be a generic point of the special fibre of the first copy of $X$ , and put $R^{'} = O_{X, ξ}$ and $K^{'} = Frac R^{'}$ . The canonical map $Spec K^{'} \to X_{K} = A$ is an $A (K^{'})$ -point $a$ coming from the first, parameter factor. Apply [F1] to $t_{a}$ and $t_{- a}$ on the second copy $X_{R^{'}}$ . Their rational-domain open immersions are inverse on fibre-dense open subsets. By finite-presentation spreading (Finite-stage descent of finitely presented schemes and their morphisms) their domains, maps and inverse identities spread over a neighbourhood of $ξ$ in the parameter $X$ . Thus $(x, y) \mapsto (x, m (x, y))$ and its inverse are defined near all special-fibre generic points of $X \times_{R} X$ projecting to $ξ$ : after localization these are generic points of the special fibre of $X_{R^{'}}$ , and the local domains are $R^{'}$ -dense. Every special-fibre component of the product projects dominantly onto a special component of the first factor, since both factors are smooth and their fibre components are geometrically regular. Repeating for its finitely many $ξ$ , and adjoining the generic translation and its inverse on the open generic fibre, gives fibre-dense domains for a rational map $Φ$ and its inverse. On overlaps the maps agree on the generic fibre and hence agree by separatedness and flatness. Restricting the two domains to where the compositions are defined gives inverse open immersions; these restrictions remain fibre-dense by the local inverse construction. Hence $Φ$ is $R$ -birational, and its second projection defines an $R$ -rational extension of $m_{K}$ .

2.1F1F2step 1.1construct

Apply the same argument with the second factor as parameter: its canonical $K^{'}$ -point, not a point of the variable first factor, supplies the translation. This constructs the other universal map $Ψ (x, y) = (m (x, y), y)$ and its rational inverse. Both are $R$ -birational on fibre-dense open domains.

3.1F2step 2.1algebra∎

The two constructions agree generically on the common dense open where both are defined, because both restrict to the generic multiplication of $A$ ; by separatedness of $X$ and schematic density of the domain ([F2]) they define a single $R$ -rational map $m$ . Associativity holds wherever the composed expressions are defined: both sides restrict to the associative law of $A$ on a schematically dense open, so by [F2] they agree; consequently $m$ is an $R$ -birational group law with birational universal translations, as claimed.

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Coherent Kunneth, the tangent bound and the proper-image dual

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field $k$ . Then:

(a) $\dim_{k} H^{1} (A, O_{A}) \leq g$ , and the tangent space of the Picard functor at the origin is $H^{1} (A, O_{A})$ ;

(b) the identity component $B = {Pic}^{0}$ is a smooth proper connected group scheme of dimension $g$ ;

(c) every ample invertible sheaf $L$ on $A$ gives a Mumford isogeny $φ_{L} : A \to B$ with finite scheme-theoretic kernel.

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ of dimension $g$ over an algebraically closed field $k$ , and an ample invertible sheaf $L$ on $A$ .

[F1]

Coherent cohomology over a field is computed by the double Cech complex of two finite separated affine covers, with the Kunneth formula and the vanishing of higher cohomology on affine opens; the cup product makes $H^{*} (A, O_{A})$ a graded commutative algebra, and the addition law makes it a connected graded Hopf algebra (Cech cohomology computes quasi-coherent cohomology on a separated scheme, Kunneth over a field, Cup-product laws, Leray spectral sequence for sheaf cohomology, Grothendieck vanishing on a Noetherian space).

[F2]

The Picard functor is represented by a separated locally finite-type group scheme with universal rigidified bundle (Picard representation by generic quotient and translates); the Mumford map and the theorem of the square are The theorem of the square and the Mumford homomorphism into the Picard group and The theorem of the cube for an abelian variety.

[F3]

Invariant differentials trivialize $Ω_{A}$ : translation identifies the cotangent space at every point with that at the identity (Differentials of a smooth morphism); the identity component of a smooth group over a perfect field is geometrically connected, regular points form a dense open, and regular equals smooth over a perfect field (Connected finite-type groups are geometrically connected, Dense regular loci on every component, Regular equals smooth over a perfect field); ample powers are very ample after a proper morphism and amplify under tensor products and pullback along finite morphisms (Every abelian variety over a field is projective, High powers of an ample line bundle embed a proper scheme, Ampleness is invariant under positive powers, Finite pullback preserves absolute ampleness, Segre embedding and its line bundle, Global functions on proper integral schemes form a finite extension of the base field).

Proof

technique · direct: bound $H^1$ by a Hopf-algebra primitivity argument, identify the Picard tangent space, then control the image of $\varphi_{\mathcal L}$ and its kernel

1.1F1givenalgebra

By [F1] compute $H^{*} (A, O_{A})$ by the double Cech complex of two finite separated affine covers: on products of affine intersections the sections are tensor products, the two augmented Cech directions compute cohomology because affine quasi-coherent higher cohomology vanishes, and field Kunneth gives $H^{*} (A \times_{k} A, O) = H^{*} (A, O) \otimes_{k} H^{*} (A, O)$ with Koszul signs. The addition law makes $H^{*} (A, O_{A})$ a connected graded Hopf algebra in which every element of $H^{1}$ is primitive. If $v_{1}, \dots, v_{r} \in H^{1}$ are linearly independent, apply the $r$ -fold coproduct to $v_{1} \dots v_{r}$ and project to $(H^{1})^{\otimes r}$ : the result is the signed sum over permutations of the independent tensors $v_{σ (1)} \otimes \dots \otimes v_{σ (r)}$ , all coefficients being $\pm 1$ , so it is nonzero even in characteristic two. Hence $H^{r} (A, O_{A}) \neq 0$ and therefore $r \leq g$ by the vanishing above dimension $g$ ; no Borel structure theorem is needed.

2.1F1F2step 1.1algebra

The exponential sequence $1 \to 1 + ε O_{A} \to O_{A [ε]}^{*} \to O_{A}^{*} \to 1$ on the dual numbers identifies the rigidified Picard tangent space with $H^{1} (A, O_{A})$ , so its dimension is at most $g$ by step 1.1.

3.1F2F3step 2.1algebra

It remains to justify finiteness of $K = \ker φ_{L}$ . The kernel is represented by the Picard scheme just constructed, and the normalized family $Λ (L)$ is trivial on $A \times_{k} K$ by the universal property of the kernel. Over the algebraic closure, $Y = K_{r e d}^{0}$ is a smooth connected proper subgroup, hence an abelian subvariety: a reduced finite-type group over a perfect field is smooth by translating its nonempty smooth locus. Restricting $Λ (L)$ to $Y \times_{k} Y$ and pulling back by $(id, - 1)$ makes $L ∣_{Y} \otimes [- 1]^{*} L ∣_{Y}$ trivial; it is ample because $L ∣_{Y}$ is ample, inversion is an automorphism and tensor products of ample sheaves are ample. A trivial ample line bundle on a proper integral variety forces dimension zero: a high power embeds it, but all sections of the trivial bundle are scalars, so the embedding is constant. Hence $\dim Y = 0$ , so $K$ is zero-dimensional proper finite type and therefore finite, including its nonreduced structure. This is the EGM argument of the cited chapter; no pre-existing ample-kernel theorem is presumed.

4.1F1F2F3step 3.1algebra∎

For an ample $L$ , the square and cube theorems [F2] with the kernel argument of step 3.1 define $φ_{L} : A \to B = {Pic}^{0}$ with finite scheme-theoretic kernel, hence image of dimension $g$ . Since $A$ is proper and $B$ separated, the image of $φ_{L}$ is closed, connected and of dimension $g$ ; at the identity $\dim O_{B, 0} \geq g$ , while its embedding dimension is bounded by $g$ by step 2.1. Thus $O_{B, 0}$ is regular of dimension $g$ , and translation makes $B$ smooth. The closed image has the same local dimension, so its defining ideal in this regular local domain is zero. It is therefore open and closed in the connected group $B$ , hence $B$ itself, and $B = {Pic}^{0}$ is a smooth proper connected group of dimension $g$ ; $φ_{L}$ is an isogeny. The identity component represents precisely algebraically trivial classes on all tests: a connected family of line bundles maps into one connected component of the Picard scheme, so differences of its fibres lie in $B$ ; conversely the universal bundle on the connected finite-type scheme $B$ connects every geometric point to the identity. Since $B$ is open, a classifying map factors through it exactly when every geometric fibre class lies there, including on nonreduced tests. Invariant differentials trivialize $Ω_{A}$ by [F3].

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Strictification

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ , let $X$ be a smooth separated faithfully flat finite-type $R$ -scheme, and let $m$ be an $R$ -birational group law on $X$ with birational universal translations (Birational group law). Then there is an $R$ -dense model open $X^{0} \subseteq X$ on which $m$ is a strict birational group law. More precisely, its multiplication is defined on an open $U^{0} \subseteq X^{0} \times_{R} X^{0}$ , and the two universal translations restrict to open immersions there whose domains and images are dense over each of the two projections. Thus every test-valued first or second coordinate gives a test-birational translation, with cancellation on arbitrary tests. The model open $X^{0}$ is smooth, separated, faithfully flat and of finite type over $R$ . If $X_{K}$ is already a group scheme and the generic law is its everywhere-defined group law, $X^{0}$ can be chosen with $X_{K}^{0} = X_{K}$ ; this includes the commissioned abelian minimal model.

Facts & Assumptions

Given: AC and DC, a DVR $R$ , a smooth separated faithfully flat finite-type $R$ -scheme $X$ , and an $R$ -birational group law $m$ with birational universal translations.

[F1]

For a quasi-compact open $V$ in a smooth $Y / S$ of relative dimension $d$ , the locus where $V$ fails to be dense in a fibre is constructible: the complement $A = Y ∖ V$ has local fibre dimension at most $d$ , the locus $F \subseteq A$ where the fibre has local dimension $d$ is closed by upper semicontinuity, and its image is constructible (Local fibre-dimension bound from polynomial quasi-finiteness, Constructible images for finite-presentation affine maps).

[F2]

If a constructible subset $C$ of a Noetherian space has nonempty irreducible closure $D$ , it contains a nonempty open of $D$ : write $C$ as a finite union of locally closed subsets $O_{j} \cap F_{j}$ in $D$ . Their closures cover $D$ , so irreducibility makes one dense; its closed part is then all of $D$ , and it contains the nonempty open $O_{j}$ . Thus a constructible subset omitting the generic point of an irreducible component has nondense closure in that component. This proves the general topological form used here; Dense constructible subsets contain an open supplies its classical-variety instance. Smooth total spaces over a DVR are regular, and their local rings at special-fibre generic points are DVRs with uniformizer $π$ (Regularity ascends and descends along a flat local homomorphism, one dimensional regular local rings are dvrs). These facts prove the special-generic closure exclusion in step 2.1; no assertion about arbitrary constructible closures over a DVR is assumed.

[F3]

$R$ -dense opens are schematically dense and behave well under base change; representatives of $S$ -rational maps agree on schematically dense opens of separated targets and descend along faithfully flat maps (S-dense open subschemes and S-rational maps, An S-rational map defined after a faithfully flat smooth base change is defined, Scheme morphisms satisfy fppf descent, Agreement on a schematically dense open).

Proof

technique · direct: remove the constructible bad-density loci for the two projections and restrict the law

1.1F1givenconstruct

Choose an $R$ -dense open $U \subseteq X^{2}$ where $m$ is defined and both universal translations $Φ (x, y) = (x, m (x, y))$ and $Ψ (x, y) = (m (x, y), y)$ are open immersions; birationality provides such a common domain by intersecting domains of the maps and their inverses. Set $V = Φ (U)$ , $W = Ψ (U)$ and $Z = U \cap V \cap W$ , all $R$ -dense. For each projection $p_{i} : X^{2} \to X$ , let $T_{i}$ be its constructible bad-density locus for $Z$ , supplied by [F1]. Every generic point of every generic or special fibre of $X$ lies outside $T_{i}$ , since $Z$ contains the generic points of all product-fibre components.

2.1F2step 1.1algebra

The closure of each $T_{i}$ omits every fibre generic point. On the generic fibre this follows from constructibility and the first assertion of [F2]. Its generic part has a reduced schematic closure whose ideal is saturated under multiplication by $π$ . At a special-fibre generic point $ξ$ , the local ring is a DVR by [F2]. A nonzero proper ideal of that DVR cannot be $π$ -saturated: divide an element $π^{n} u$ repeatedly to obtain a unit. The localized closure ideal is nonzero because the generic bad locus is nondense in its integral component. Hence this generic-part closure misses $ξ$ . The special part is constructible within $X_{k}$ and omits its generic points; [F2] makes its closure nondense there as well. Thus $Q_{i} = X ∖ \overline{T_{i}}$ is an $R$ -dense open. Set $X^{0} = Q_{1} \cap Q_{2}$ ; over it $Z$ is dense along both projections.

3.1F1F2step 2.1algebra

Define $U^{0} = U \cap (X^{0} \times_{R} X^{0}) \cap m^{- 1} (X^{0})$ , with images $V^{0} = Φ (U^{0})$ and $W^{0} = Ψ (U^{0})$ in $(X^{0})^{2}$ . To check density, base change to a field and fix a point $a \in X^{0}$ . The translation $Φ (a, -)$ is an open immersion with dense image in the fibre; intersecting that image with $V \cap (a \times X^{0})$ remains dense. Its inverse image imposes $m (a, -) \in X^{0}$ , so $U^{0}$ is dense along $p_{1}$ . The same argument with $Ψ (-, a)$ proves density along $p_{2}$ . Since $Φ$ preserves $p_{1}$ and $Ψ$ preserves $p_{2}$ , it also proves $V^{0}$ dense along $p_{1}$ and $W^{0}$ along $p_{2}$ .

4.1F1step 3.1algebra

For the remaining densities fix $a \in X^{0}$ and put $U_{a} = m^{- 1} (a) \subseteq U$ over the chosen field. The open immersions $Φ$ and $Ψ$ identify $U_{a}$ respectively with $V \cap (X \times a)$ and $W \cap (a \times X)$ , dense opens by the construction of $X^{0}$ . Requiring both input coordinates to lie in $X^{0}$ cuts two dense opens of $U_{a}$ , so $U_{a} \cap U^{0}$ is dense in $U_{a}$ . Its $Ψ$ -image is $W^{0} \cap (a \times X^{0})$ , dense along the first projection; its $Φ$ -image is $V^{0} \cap (X^{0} \times a)$ , dense along the second. All domains and images therefore have both-projection density.

5.1F3step 2.1step 4.1algebra∎

The restricted rational law on $X^{0}$ is associative by its agreement with the original law on schematic dense domains. Its universal translations are the open immersions above, and both-projection density remains schematic density after arbitrary coordinate base change in the smooth family by [F3]. Thus they give the required test-birational translations and cancellation. The open $X^{0}$ is smooth, separated and finite type; it meets every special-fibre component and the generic fibre, so it is surjective and flat over $R$ , hence faithfully flat. If $X_{K}$ already carries an everywhere-defined group law, choose $U$ to include $(X_{K})^{2}$ , where both universal translations are isomorphisms. The generic bad loci are then empty, so $X_{K}^{0} = X_{K}$ . This is the BLR model-open strictification needed by completion.

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Finite-field descent of the dual and the Poincare bundle

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be an abelian variety over a field $k$ . Then the algebraically trivial rigidified Picard subfunctor of $A / k$ is represented by an abelian variety $A^{\lor}$ of dimension $\dim A$ , together with a normalized Poincare bundle $P$ on $A \times_{k} A^{\lor}$ , and the formation and full universal property are compatible with field extension.

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ over a field $k$ .

[F1]

Over an algebraically closed field the entire rigidified Picard functor is represented by a separated locally finite-type group scheme (Picard representation by generic quotient and translates); its identity component is smooth proper of dimension $\dim A$ (Coherent Kunneth, the tangent bound and the proper-image dual); the universal rigidified bundle is obtained by evaluation at the identity map of the representative, and its restriction to $A \times B$ normalized on both axes gives the Poincare bundle (Rigidification and effective descent of line bundles).

[F2]

Finite data spread from the algebraic closure to a finite extension: objects and morphisms of finite presentation descend along filtered colimits; morphisms and isomorphisms of finitely presented line bundles also descend to finite stages, and compatible morphisms descend along finite faithfully flat field extensions; and a scheme projective over a finite field extension is projective over the ground field (Finite-stage descent of finitely presented schemes and their morphisms, Finite-stage descent of finitely presented quasi-coherent sheaves, Scheme morphisms satisfy fppf descent, Faithfully flat descent of modules and algebras is effective).

[F3]

The affine-orbit descent for a finite-field extension applies to a smooth proper connected group scheme whose finite descent orbits lie in affine opens, and the orbit condition follows from Serre vanishing for a high power of a very ample line bundle and sections avoiding finitely many closed specializations (Finite field descent is effective for schemes with affine-contained descent orbits, Every abelian variety over a field is projective, High powers of an ample line bundle embed a proper scheme, Ampleness is invariant under positive powers, Finite pullback preserves absolute ampleness, Projective coherent finiteness and large twist vanishing, Global functions on proper integral schemes form a finite extension of the base field, Segre embedding and its line bundle).

Proof

technique · direct: construct over the algebraic closure, spread the finite data to a finite extension, then descend the representative

1.1F1givenconstruct

Work over an algebraic closure and write $G$ for the represented full rigidified Picard functor and $B = G^{0}$ for its identity component. By [F1], $B$ is smooth proper connected of dimension $\dim A$ , hence an abelian variety and projective. We establish the algebraically trivial identification directly. The connected components of the locally finite-type scheme $G$ are open and closed, and translation identifies them with cosets of $B$ . A rigidified line-bundle family on a connected finite-type parameter scheme gives a morphism to $G$ , whose image lies in one connected component; therefore differences of its geometric fibre classes lie in $B$ . A chain of such differences has the same property. Conversely, restricting the universal rigidified bundle on $A \times G$ to $A \times B$ gives a connected finite-type family whose identity fibre is trivial and whose fibre at any geometric point $b$ has class $b$ , proving that every $B$ -class is algebraically trivial. For an arbitrary test scheme $T$ , its classifying morphism $T \to G$ factors through the open subscheme $B$ exactly when all geometric fibre classes lie in $B$ : the inverse image of the complementary open-and-closed components is empty if it has no geometric point. This argument also retains every nilpotent of $T$ , since factorization through an open imposes no reduction. Thus $B$ represents the entire algebraically trivial rigidified subfunctor on all tests. Restricting the universal bundle and normalizing on both axes now gives the Poincare bundle.

2.1F1F2step 1.1algebra

For arbitrary $k$ , spread $B$ , a projective embedding, its group operations and the rigidified Poincare bundle from $\overset{ˉ}{k}$ to a finite extension $K / k$ by [F2]. The resulting natural transformation to the algebraically trivial rigidified functor is an isomorphism after base change to $\overset{ˉ}{k}$ . For an affine $K$ -test $T$ and a rigidified algebraically trivial bundle on $A_{T}$ , its unique classifying morphism over $T_{\overset{ˉ}{k}}$ and the isomorphism with the pulled-back Poincare bundle descend to $T_{K^{'}}$ for some finite extension $K^{'} / K$ inside $\overset{ˉ}{k}$ , by the finite-presentation statements of [F2]. Uniqueness is detected after faithful field extension: two classifying morphisms for the same bundle become equal over $\overset{ˉ}{k}$ by [F1], hence were equal already. This also applies on the finite cover's double overlap, including its nilpotents, by tensoring that overlap with $\overset{ˉ}{k}$ over $K$ and using the all-test universal property over $\overset{ˉ}{k}$ . Thus the local classifying morphism has equal pullbacks and descends along the finite fppf cover $T_{K^{'}} \to T$ by [F2]; the bundle isomorphism descends by module descent and rigidity. Affine-test extensions glue uniquely, proving the universal property over $K$ on every test scheme.

3.1F2F3step 2.1construct

Consequently $B_{K}$ carries a canonical descent datum over $K \otimes_{k} K$ from the uniqueness of representatives of the same base-changed functor; the datum includes the entire nonreduced tensor algebra and satisfies the cocycle by uniqueness. The scheme $B_{K}$ is smooth, proper, connected, projective over $K$ , hence as a $k$ -scheme projective too, since $Spec K \to Spec k$ is finite and projective. Every finite descent orbit lies in an affine open of this projective scheme: choose a closed specialization of each of its finitely many points; for a high power of a very ample line, Serre vanishing makes the map to the fibres at those finitely many closed points surjective, and a section nonzero at each of them has an affine nonvanishing locus; nonvanishing at a specialization implies nonvanishing at the original point. The exact finite-field descent lemma [F3] now descends $B_{K}$ , including inseparable $K$ .

4.1F2step 3.1algebra∎

Module descent gives the Poincare bundle $P$ , compatible morphism descent gives the group law and the rigidifications, and the sheaf isomorphism descends, proving the full universal property over $k$ and its compatibility with field extension.

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Strict law graph calculus

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring, $X$ a smooth separated faithfully flat finite-type $R$ -scheme, and $m$ a strict $R$ -birational group law on $X$ (Strictification). Then:

(a) $X$ embeds into the functor of relative birational self-maps of $X$ , and the closure $Γ$ of the multiplication graph in $X \times_{R} X \times_{R} X$ has all three two-coordinate projections which are open immersions with both-projection dense images;

(b) for a section $a$ and a point $b$ , the section translation $t_{a}$ is defined at $b$ if and only if the law $m$ is defined at $(a, b)$ ;

(c) products of section translations are computed by the graph triple: if the law is defined at the relevant pairs, then $t_{a} t_{b} = t_{c}$ where $c$ is the third coordinate of the graph.

Facts & Assumptions

Given: AC and DC, a DVR $R$ , a smooth separated faithfully flat finite-type $R$ -scheme $X$ , and a strict $R$ -birational group law $m$ on $X$ .

[F1]

Strictness: on an open $U \subseteq X \times_{R} X$ dense over both projections, the universal left and right translations are open immersions with both-projection dense images; the law is associative as an $R$ -rational map (Strictification).

[F2]

The graph of a rational map has a schematic closure containing it as a schematically dense open; since the graph domain is smooth and reduced, the closure is reduced. Schematic density survives product with the flat $R$ -scheme $X$ . Fibre-dense opens in smooth finite-presentation schemes remain schematically dense after arbitrary test-scheme base change, and two morphisms into a separated target which agree on a schematically dense open agree everywhere (Schematic closure and agreement on a dense open, Projective weak models and rational mapping, Agreement on a schematically dense open).

[F3]

A finite-type morphism with at most one point in each geometric fibre is quasi-finite. A separated quasi-finite birational morphism from a reduced source to a normal target is an open immersion componentwise: apply Zariski Main locally, then use that a finite birational algebra inside the target's fraction field equals the normal domain (Scheme Zariski Main factorization for separated quasi-finite morphisms, Regularity ascends and descends along a flat local homomorphism, Locally standard smooth iff flat with geometrically regular fibres, regular local rings are normal).

[F4]

Pullback of a faithfully flat morphism is faithfully flat and detects equality of morphisms: if two maps become equal after pullback along a faithfully flat cover, they were equal before pullback (Faithfully flat scheme morphism).

Proof

technique · follow BLR 5.2/4 and 5.3/1–4: represent elements by strict translations, prove the graph relation on a dense auxiliary variable, then use normality and Zariski Main

1.1F1F2F4givenconstruct

For an $R$ -scheme $T$ , let ${B i r}_{X / R} (T)$ be the group of $T$ -birational self-maps of $X_{T} = X \times_{R} T$ . Strictness [F1] makes each section $a \in X (T)$ act by a $T$ -birational left translation $τ_{a}$ , naturally in $T$ . This defines $X \to {B i r}_{X / R}$ . It is a monomorphism: if $τ_{a} = τ_{b}$ , then on the common dense domain the maps $(τ_{a}, id)$ and $(τ_{b}, id)$ from $T \times_{R} X$ to $X_{T} \times_{T} X_{T}$ agree. They factor as the universal right translation $(x, y) \mapsto (m (x, y), y)$ after $(a, id)$ and $(b, id)$ , respectively. The right translation is an open immersion by [F1], so cancellation gives $(a, id) = (b, id)$ on that dense open; [F2] makes the open schematically dense, hence the equality holds on $T \times_{R} X$ . Since $T \times_{R} X \to T$ is faithfully flat, [F4] gives $a = b$ . Associativity also gives $τ_{a} τ_{b} = τ_{c}$ whenever $m (a, b) = c$ is defined.

2.1F1F2step 1.1algebra

Let $Γ \subseteq X^{3}$ be the schematic closure of the graph of $m ∣_{U}$ , with coordinates $(x, y, z)$ . On the open locus in $X^{4}$ where $(y, w), (x, m (y, w)), (z, w) \in U$ , the maps $m (x, m (y, w))$ and $m (z, w)$ are defined. This locus is dense over the first three coordinates by the two-projection density of $U$ and the open-immersion property of the strict translations in [F1]. On the intersection with the graph over $U$ , associativity makes the two morphisms agree on the dense open where the associative identity is represented; as the target $X$ is separated, [F2] gives equality on their common domain. The graph over $U$ , after product with the flat scheme $X$ , is schematically dense in $Γ \times_{R} X$ ; hence [F2] extends this equality to the whole common domain in $Γ \times_{R} X$ . Pulling back along any $T$ -valued triple $(a, b, c) : T \to Γ$ shows $τ_{a} τ_{b} = τ_{c}$ as $T$ -birational maps. In particular, if $m (a, b) = c$ is defined, then $t_{a} t_{b} = t_{c}$ , proving (c); conversely, for fixed any two coordinates of a triple in $Γ (T)$ , the third is unique by the monomorphism of step 1.1 and invertibility in ${B i r}_{X / R} (T)$ .

3.1F1F3step 2.1construct

Each projection $q_{i j} : Γ \to X^{2}$ is a monomorphism: for every $T$ , a triple $(a, b, c) \in Γ (T)$ satisfies the translation relation of step 2.1, so any two of its coordinates determine the third. It is finite type, hence quasi-finite by [F3]. On the graph over $U$ , $q_{12}$ is the identity onto $U$ , while $q_{13}$ and $q_{23}$ are the universal left and right translations; these are open immersions with dense images by [F1]. Thus each $q_{i j}$ is birational on every component it meets. The target $X^{2}$ is normal because it is smooth over the regular DVR $R$ . Applying [F3] componentwise shows each $q_{i j}$ is an open immersion. Its image contains respectively $U$ , the left-translation image, and the right-translation image, all dense over both projections, so the images are dense over both projections. This proves (a).

4.1F1F2F3step 3.1step 2.1construct∎

The open immersion $q_{12}$ identifies $Γ$ with an open $W \subseteq X^{2}$ , and the third coordinate defines $m$ on $W$ . This is the full domain: any local morphism extending $m$ has graph in the closed $Γ$ , since it agrees with the graph over $U$ on a schematically dense open. For a section $a : T \to X$ and a $T$ -point $b$ of $X_{T}$ , if $m$ is defined at $(a, b)$ , pullback along $a \times id$ shows $t_{a}$ is defined at $b$ . Conversely, if $t_{a}$ is defined at $b$ , choose an open neighbourhood $D$ on which it is a morphism and consider $D \to X^{3}$ , $y \mapsto (a (p_{T} (y)), y, t_{a} (y))$ , where $p_{T} : X_{T} \to T$ . On the T-dense open where the strict law defines $t_{a}$ , this graph factors through $Γ$ ; universal schematic density [F2] therefore makes it factor through $Γ$ on $D$ . Hence $(a, b) \in W$ , so the law is defined there. This proves (b) for every test scheme. For an $R$ -section $a$ , the graph closure $Γ_{a} \subseteq X^{2}$ of $t_{a}$ maps into $Γ$ . Its two projections are finite-type monomorphisms by the monomorphism of $q_{12}$ and $q_{13}$ , and are birational because $t_{a}$ is an $R$ -birational map. The target $X$ is normal; [F3] makes both projections open immersions with dense images, as used for translate gluing.

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Homogeneous bundles and Mumford surjectivity

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be an abelian variety over a field $k$ and let $M$ be an invertible sheaf on $A$ . Then the Mumford homomorphism $φ_{M}$ (Coherent Kunneth, the tangent bound and the proper-image dual) is zero exactly when the class of $M$ lies in the connected component ${Pic}^{0}$ ; if $M$ is nontrivial and homogeneous, then $H^{i} (A, M) = 0$ for every $i$ . Over an algebraic closure every homogeneous invertible sheaf is of the form $t_{x}^{*} L \otimes L^{- 1}$ for some $x$ and a fixed ample $L$ ; the equality $\ker φ = {Pic}^{0}$ holds as a sheaf on all tests.

Facts & Assumptions

Given: AC and DC, an abelian variety $A / k$ , an invertible sheaf $M$ on $A$ , and a fixed ample invertible sheaf $L$ .

[F1]

Over an algebraic closure the entire rigidified Picard functor is represented on all tests by Picard representation by generic quotient and translates, while its identity component and the dual/Poincare bundle are supplied by Coherent Kunneth, the tangent bound and the proper-image dual and Finite-field descent of the dual and the Poincare bundle. The field-level square homomorphism and cube identity are The theorem of the square and the Mumford homomorphism into the Picard group and The theorem of the cube for an abelian variety. For any test family $M$ the normalized square $Λ (M) = m^{*} M \otimes p_{1}^{*} M^{- 1} \otimes p_{2}^{*} M^{- 1} \otimes π^{*} e^{*} M$ defines a morphism $φ_{M} : A_{T} \to A_{T}^{\lor}$ : its fibre classes are translation differences, hence algebraically trivial, and it is rigidified on both axes. Uniqueness and effective descent of rigidified bundles are Rigidification and effective descent of line bundles.

[F2]

Cohomology of coherent sheaves on $A$ is computed by Cech complexes with Kunneth and Leray techniques, and the rigid-factor lemma applies to morphisms from products with an abelian factor (Cech cohomology computes quasi-coherent cohomology on a separated scheme, Kunneth over a field, Leray spectral sequence for sheaf cohomology, Universal finite projective cohomology complex over any base, Cohomology and base change for proper flat coherent families, Grothendieck vanishing on a Noetherian space, Rigidity for a proper geometrically integral factor).

[F3]

Proper geometrically integral schemes have only scalar global functions (Global functions on proper integral schemes form a finite extension of the base field). For ample $L$ , $φ_{L}$ has finite scheme-theoretic kernel over an algebraic closure (Coherent Kunneth, the tangent bound and the proper-image dual).

Proof

technique · direct: construct the normalized-square map and use rigidity for the Picard identity component, then prove homogeneous vanishing and surjectivity by Kunneth and the two Leray sequences

1.1F1F2givenconstruct

Work first over an algebraic closure. The Poincare family on $A \times B$ , where $B = {Pic}^{0} = A^{\lor}$ , gives through [F1] a morphism $A \times B \to B$ ; it is zero on $A \times {0}$ and on ${0} \times B$ . The proper-factor rigidity lemma [F2] makes it zero everywhere, as an identity of morphisms. Thus every family classified by $B$ has zero Mumford map, even on nonreduced tests. The normalized square defines the map for any bundle as in [F1]. For a bundle over the ground field it is a homomorphism: the square identity establishes addition on geometric points, and the two resulting morphisms from the reduced $A \times A$ to separated $B$ therefore agree. For a test family, locally its classifying map lands in a component of the full Picard scheme. Each component is a translate of $B$ , and its universal family is a fixed bundle tensored with the Poincare family. Tensor product adds normalized-square maps, so the preceding vanishing makes this family map the base change of the fixed bundle's homomorphism. Consequently the construction gives homomorphisms on all tests and commutes with base change.

2.1F1F2F3step 1.1algebra

Let $M$ be a ground-field bundle with $φ_{M} = 0$ ; by the normalized-square universal property its square family is trivial, giving $m^{*} M ≅ p_{1}^{*} M \otimes p_{2}^{*} M$ after trivializing the constant identity fibre. Pulling back along $(id, - 1)$ gives $[- 1]^{*} M ≅ M^{- 1}$ . If $M$ has a nonzero section, inversion gives a nonzero section of $M^{- 1}$ ; their product is a nonzero scalar by integrality and [F3], so $M$ is trivial. A nontrivial $M$ therefore has $H^{0} (M) = 0$ . If $i > 0$ is the least degree with $H^{i} (M) \neq 0$ , multiplication pullback followed by restriction along $(id, 0)$ is the identity on $H^{i} (M)$ , but Kunneth identifies the intermediate group with $⨁_{a + b = i} H^{a} (M) \otimes H^{b} (M) = 0$ . This contradiction proves vanishing in every degree. Flat field base change gives the same vanishing over the original field.

3.1F1F2F3step 1.1step 2.1algebra

Over an algebraic closure suppose $M$ has zero Mumford map but is not $t_{x}^{*} L \otimes L^{- 1}$ for any $x$ , and put $Q = Λ (L) \otimes p_{2}^{*} M^{- 1}$ . On a $p_{1}$ -fibre it is the nontrivial bundle $t_{x}^{*} L \otimes L^{- 1} \otimes M^{- 1}$ , whose Mumford map is zero by step 1.1. Step 2.1 and the universal cohomology complex give $R p_{1, *} Q = 0$ , hence $H^{*} (Q) = 0$ . On a $p_{2}$ -fibre its class is $t_{y}^{*} L \otimes L^{- 1}$ , since the other factors are constant lines; it has zero cohomology away from the finite kernel $K (L)$ supplied by [F3]. All $R^{i} p_{2, *} Q$ thus have finite support. They have no higher cohomology, so the second Leray sequence identifies their global sections with $H^{i} (Q) = 0$ and makes every direct image zero. Derived base change then makes every fibre cohomology zero, contradicting the trivial bundle on the fibre at $y = 0$ . Therefore every such $M$ is a Mumford translate for the fixed ample $L$ .

4.1F1step 1.1step 3.1algebra∎

A zero-Mumford test family has, by step 3.1 on each geometric fibre, all its fibre classes in the open identity component $B$ of the represented Picard scheme. Its classifying map therefore factors through $B$ , including its nilpotent structure; no reduced-test argument is used. Conversely, step 1.1 makes every $B$ -classified family have zero Mumford morphism. Thus the kernel sheaf is exactly ${Pic}^{0}$ on all tests over an algebraic closure. Rigidified bundle descent and the field-compatible dual of [F1] descend this equality to $k$ . This proves the asserted criterion, vanishing and geometric surjectivity.

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Separated translate gluing

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a strictly henselian discrete valuation ring with fraction field $K$ and residue field $k$ , let $X$ be a smooth separated finite-type $R$ -scheme with a strict $R$ -birational group law $m$ , and let $a$ be an $R$ -section of $X$ . Then gluing $X$ to a left translate $X (a)$ along the closed section-translation graph produces a smooth separated finite-type $R$ -scheme $X^{'}$ containing $X$ as an $R$ -dense open subscheme and extending the strict law $m$ to a strict law on $X^{'}$ .

Facts & Assumptions

Given: AC and DC, a strictly henselian discrete valuation ring $R$ , a smooth separated finite-type $R$ -scheme $X$ with a strict birational group law $m$ , and a section $a : Spec R \to X$ .

[F1]

For a strict law the graph closure of the section translation has two-coordinate projections that are open immersions with dense images, and section translation is defined at $b$ exactly when the law is defined at $(a, b)$ (Strict law graph calculus).

[F2]

Gluing along open subschemes is available, and rational maps agree when they agree on schematically dense opens of separated reduced targets and descend along faithfully flat maps (Gluing affine schemes along compatible open isomorphisms, S-dense open subschemes and S-rational maps, An S-rational map defined after a faithfully flat smooth base change is defined, Scheme morphisms satisfy fppf descent, Agreement on a schematically dense open).

[F3]

A separated quasi-finite birational morphism with integral source and normal target is an open immersion, by the finite-birational component argument in Scheme Zariski Main factorization for separated quasi-finite morphisms. Smooth schemes over a DVR are regular and normal, as established in [F1].

Proof

technique · direct: glue along the translation graph and check the strict-law conditions on the pieces

1.1F1F2givenconstruct

Let $Γ \subseteq X \times_{R} X$ be the graph closure of the section translation $t_{a}$ ; by [F1] its two projections are open immersions onto $R$ -dense open subschemes. Gluing $X$ to a copy $X (a)$ along $Γ$ is therefore gluing along an open subscheme, giving a smooth finite-type $R$ -scheme $X^{'}$ containing $X$ as an $R$ -dense open subscheme; the closedness of the graph in the separated product makes $X^{'}$ separated.

2.1F1F2step 1.1construct

Write $j : X \tilde{\to} X (a)$ for the canonical copy map, which extends the left translation by $a$ on its original domain. Let $U$ be the strict-law domain in $X^{2}$ , with translation images $V$ and $W$ . On $U_{1} = (j \times id) (U)$ define $m^{'} (j (x), y) = j (m (x, y))$ . On $U_{2} \subset X \times X (a)$ , consisting of $(x, j (y))$ with $(x, a) \in U$ and $(m (x, a), y) \in U$ , define $m^{'} (x, j (y)) = m (m (x, a), y)$ . Together with $m$ on $U$ these morphisms agree on overlaps by associativity and schematic density [F2], and give $m^{'}$ on $U^{'} = U \cup U_{1} \cup U_{2}$ . For any fixed $y$ , strictness makes the conditions on $x$ in $U_{2}$ dense: right translation by $a$ is birational, and the domain of right translation by $y$ is dense. Thus $U_{2}$ is dense along its second projection. The domains $U$ and $U_{1}$ are dense along the first projection; since $X$ is fibre-dense in $X^{'}$ , these facts make $U^{'}$ dense along both projections of $(X^{'})^{2}$ . No definition on the whole fourth chart $X (a)^{2}$ is needed for this density assertion.

3.1F1F2F3step 2.1algebra∎

The universal translations on each of $U, U_{1}, U_{2}$ are open immersions: on $U_{1}$ conjugate the original translations by the copy isomorphism $j$ ; on $U_{2}$ compose the original translations with the open-immersion right translation by $a$ and with $j$ . Hence the glued translations are quasi-finite. They are birational and separated, and their source and target are regular componentwise; [F3] makes them open immersions. The left-translation image contains $V$ and $(j \times j) (V)$ , so it is dense along both projections. The right-translation image contains $W$ and $(j \times id) (W)$ , giving first-projection density and second-projection density over $X$ . Over a second coordinate $j (y)$ , its image from $U_{2}$ is the set of products $(x a) y$ on the dense domain just described; the composite of the birational right translations by $a$ and $y$ has dense image in $X$ , hence in $X^{'}$ . This gives second-projection density over $X (a)$ as well. Thus $m^{'}$ is strict. Associativity follows from its agreement with $m$ on schematically dense domains.

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Dual isogenies, Cartier-dual kernels and canonical biduality

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ and $C$ be abelian varieties over a field $k$ and let $f : A \to C$ be an isogeny, that is, a surjective homomorphism whose scheme-theoretic kernel $H = \ker f$ is finite; write $\deg f = rank H = \dim_{k} O (H)$ , so $f$ is finite flat of degree $\deg f$ . Let $A^{\lor}, C^{\lor}$ be the dual abelian varieties with normalized Poincare bundles (Finite-field descent of the dual and the Poincare bundle, The rigidified relative Picard functor and the dual abelian variety). Then:

(a) [dual isogeny] the rule $N \mapsto f_{T}^{*} N$ on $T$ -points defines a homomorphism $f^{\lor} : C^{\lor} \to A^{\lor}$ of abelian varieties, and $f \mapsto f^{\lor}$ is contravariantly functorial in $f$ ;

(b) [kernel] there is an isomorphism of $k$ -group schemes $\ker f^{\lor} ≅ H^{D}$ onto the Cartier dual of $H$ (Finite Cartier duality, exactness and exponent);

(d) [biduality and functoriality] the canonical morphism $κ_{A} : A \to A^{\lor \lor}$ given on $T$ -points by the class of the switched normalized Poincare bundle is an isomorphism. For composable homomorphisms $f : A \to C$ and $g : C \to D$ , duality satisfies $(id)^{\lor} = id$ , $(g \circ f)^{\lor} = f^{\lor} \circ g^{\lor}$ , and $f^{\lor \lor} \circ κ_{A} = κ_{C} \circ f$ . It is additive: for homomorphisms $f, g : A \to C$ with the same source and target, $(f + g)^{\lor} = f^{\lor} + g^{\lor}$ . The kernel and degree assertions in (b) concern isogenies.

Facts & Assumptions

Given: AC and DC, abelian varieties $A, C$ over a field $k$ , an isogeny $f : A \to C$ with kernel $H$ and degree $d = \dim_{k} O (H)$ , and the dual abelian varieties $A^{\lor}$ , $C^{\lor}$ with normalized Poincare bundles.

[F1]

The dual abelian variety represents the degree-zero rigidified relative Picard functor on all $k$ -schemes, with normalized Poincare bundle $P$ on $A \times_{k} A^{\lor}$ , and the formation is compatible with field extension; moreover the rigidified relative Picard functor is an fppf sheaf and rigidified line bundles have no nontrivial automorphisms (Finite-field descent of the dual and the Poincare bundle, Rigidification and effective descent of line bundles, The rigidified relative Picard functor and the dual abelian variety).

[F2]

For every invertible sheaf $M$ on an abelian variety and every test scheme the Mumford homomorphism $φ_{M}$ vanishes exactly when the class of $M$ lies in ${Pic}^{0}$ ; when $M$ is normalized along the identity and $φ_{M} = 0$ , one has $m^{*} M ≅ p_{1}^{*} M \otimes p_{2}^{*} M$ , and the Mumford map is compatible with pullback along homomorphisms: $φ_{f^{*} M} = f^{*} \circ φ_{M} \circ f$ (Homogeneous bundles and Mumford surjectivity, The theorem of the square and the Mumford homomorphism into the Picard group). Every abelian variety is projective and hence admits an ample invertible sheaf (Every abelian variety over a field is projective).

[F3]

Finite Cartier duality is an exact contravariant equivalence on finite commutative $k$ -group schemes, $H^{D}$ of rank $d$ represents the all-test characters $T \mapsto {Hom}_{T -groups} (H_{T}, G_{m, T})$ , and $H$ is killed by $d$ (Finite Cartier duality, exactness and exponent).

[F4]

The quotient $A / H$ exists as a separated finite-type $k$ -group scheme, the projection $A \to A / H$ is faithfully flat of finite presentation with scheme-theoretic kernel $H$ and is an $H$ -torsor; any homomorphism of finite-type $k$ -group schemes with trivial kernel is a closed immersion (Normal subgroup quotients of finite-type group schemes exist as fppf scheme quotients, Finite-type algebraic group monomorphisms are closed immersions).

[F5]

Modules and commutative algebras with descent data along a faithfully flat map are effectively descended, the descent equivalence is monoidal, and finitely generated locally free modules are detected after faithfully flat base change (Faithfully flat descent of modules and algebras is effective).

[F6]

For every abelian variety (in particular $A$ , $C$ and their duals) and every base change $A_{T} \to T$ the unit map is an isomorphism; in particular $p_{T, *} O_{A_{T}} = O_{T}$ , so every unit on $A_{T}$ is a unit pulled back from $T$ (Universal structure-sheaf sections of an abelian scheme). The theorem of the cube holds for abelian varieties, and multiplication is finite faithfully flat of rank $∣ n ∣^{2 g}$ with finite locally free kernel (The theorem of the cube for an abelian variety, Nonzero multiplication on an abelian variety is finite and faithfully flat).

Proof

technique · direct: realize $f$ as the fppf quotient by $H$, describe the kernel of the pullback rule by equivariant descent and Cartier characters, then compare the two Mumford isogenies through the canonical biduality morphism

1.1F4givenalgebra

The kernel $H$ is a finite $k$ -group scheme, and since $A / H$ exists and the homomorphism $A \to A / H$ has kernel $H$ , the induced morphism $A / H \to C$ is a closed immersion [F4] which is surjective because $f$ is surjective; since $C$ is reduced, a surjective closed immersion into $C$ has zero defining ideal and is therefore an isomorphism, so $f$ is the quotient map $A \to A / H$ and in particular a faithfully flat $H$ -torsor of finite presentation of degree $d = rank H$ .

1.2F1F2givenconstruct

For a $k$ -scheme $T$ and a rigidified line bundle $N$ on $C_{T}$ define $f_{T}^{\lor} (N) = f_{T}^{*} N$ ; this is a rigidified line bundle on $A_{T}$ (pullback of the rigidification), and it is degree zero: for $N \in C^{\lor} (T)$ one has $φ_{N} = 0$ , so by [F2] $φ_{f_{T}^{*} N} = f_{T}^{*} \circ φ_{N} \circ f_{T} = 0$ , hence $f_{T}^{*} N \in {Pic}^{0} (A_{T}) = A^{\lor} (T)$ . The rule is compatible with base change in $T$ and with tensor products, and pullback of bundles is contravariant, so $f \mapsto f^{\lor}$ is a contravariant additive functor; since source and target are represented by $C^{\lor}$ and $A^{\lor}$ , Yoneda's lemma promotes $f^{\lor}$ to a homomorphism of $k$ -group schemes.

1.3F1F6givenconstruct

The canonical morphism $κ_{A} : A \to A^{\lor \lor}$ is defined by the switched normalized Poincare bundle: for a test $T$ , the pullback $Q = (swap)^{*} P$ is a rigidified line bundle on $A^{\lor} \times_{k} A$ which is degree zero along $A^{\lor}$ , hence it represents a morphism $κ_{A} : A \to A^{\lor \lor} = (A^{\lor})^{\lor}$ by the universal property [F1], and $κ_{A}$ is a homomorphism of group schemes because the biextension identities of $P$ are multiplicative in the second variable, which is the theorem of the cube [F6].

2.1F3F4F5F6step 1.1construct

We compute $\ker f^{\lor}$ . Let $T$ be a $k$ -scheme and let $N \in \ker f^{\lor} (T)$ , so there is an isomorphism $α : f_{T}^{*} N \to O_{A_{T}}$ of rigidified line bundles. Since $f_{T} : A_{T} \to C_{T}$ is an $H_{T}$ -torsor [step 1.1], the pair $(N, α)$ is exactly a descent datum for the trivial line bundle $O_{A_{T}}$ along $f_{T}$ : an isomorphism $θ : p_{1}^{*} O \to p_{2}^{*} O$ over $A_{T} \times_{C_{T}} A_{T} ≅ A_{T} \times_{T} H_{T}$ satisfying the cocycle condition. Write $θ (h) \in H^{0} (A_{T}, O^{*}) = O_{T}^{*}$ for the unit attached to $h \in H_{T}$ ; by [F6] every such unit is pulled back from $T$ . The cocycle condition becomes $θ (h_{1} h_{2}) = θ (h_{1}) θ (h_{2})$ in $O_{T}^{*}$ , so $θ$ is precisely an $H_{T}$ -valued character, that is, an element of ${Hom}_{T -gr} (H_{T}, G_{m, T}) = H^{D} (T)$ by [F3].

3.1F1F2F3F5F6step 2.1algebra

The character in step 2.1 is independent of the chosen trivialization: changing it by a base unit conjugates the scalar action trivially. Conversely, a character $χ \in H^{D} (T)$ gives an $H_{T}$ -linearization of $O_{A_{T}}$ ; monoidal effective descent [F5] gives a line bundle $N$ on $C_{T}$ with $f_{T}^{*} N ≅ O_{A_{T}}$ and an induced rigidification. It remains to place $N$ in $C^{\lor} (T)$ on the entire test scheme. Since $[d]_{H} = 0$ by [F3], $χ^{d} = 1$ , so its $d$ -th tensor-power descent datum is trivial and $N^{d}$ is rigidified-trivial. Multiplicativity of the normalized square family gives $d φ_{N} = φ_{N^{d}} = 0$ by [F2]. Thus the pointed morphism $φ_{N} : C_{T} \to C_{T}^{\lor}$ factors through the finite affine $T$ -group $C^{\lor} [d]_{T}$ supplied by [F6]. The universal structure-sheaf equality for $C_{T}$ in [F6] identifies maps to this relative affine target with algebra maps to $O_{T}$ , so the morphism factors through $T$ ; evaluation at the identity makes that factor the zero section. Hence $φ_{N} = 0$ on all tests, and [F2] gives $N \in {Pic}^{0} (C_{T}) = C^{\lor} (T)$ . The constructions are inverse and natural, and tensor products agree with products of characters. Therefore $\ker f^{\lor} ≅ H^{D}$ as fppf group sheaves, and hence as group schemes by representability.

4.1F3step 1.2step 3.1algebra

Consequently $f^{\lor}$ is finite (its kernel $H^{D}$ is finite) and $\dim C^{\lor} = \dim C = \dim A = \dim A^{\lor}$ , so its image is a closed connected subgroup of dimension $\dim A^{\lor}$ , hence all of $A^{\lor}$ ; thus $f^{\lor}$ is an isogeny. Applying the quotient-torsor argument of step 1.1 to it proves finite flatness, with $\deg f^{\lor} = rank \ker f^{\lor} = rank H^{D} = d = \deg f$ , using that Cartier duality preserves ranks [F3].

5.1F1F2F6step 4.1algebra

To prove that $κ_{A}$ is an isogeny, choose an ample line bundle $L$ on $A$ and put $B = A^{\lor}$ , $f = φ_{L} : A \to B$ , an isogeny by [F2]. Let $P_{A}$ and $P_{B}$ be the normalized Poincare bundles of $A$ and $B$ . By the definition of $κ_{A}$ , $({id}_{B} \times κ_{A})^{*} P_{B} ≅ {swap}^{*} P_{A}$ on $B \times A$ . The defining dual-pullback identity also gives $(f \times {id}_{B^{\lor}})^{*} P_{B} ≅ ({id}_{A} \times f^{\lor})^{*} P_{A}$ . Pulling these identities to $A \times A$ identifies $({id}_{A} \times f^{\lor} κ_{A})^{*} P_{A}$ with ${swap}^{*} ({id}_{A} \times f)^{*} P_{A} = {swap}^{*} Λ (L)$ . The normalized bundle $Λ (L)$ is symmetric, so the universal property of $P_{A}$ yields $f = f^{\lor} \circ κ_{A}$ . Every pullback in this comparison has the displayed product domain; in particular $({id}_{A} \times φ_{L})^{*} P_{A}$ lies on $A \times A$ . Since $f$ has finite kernel, $κ_{A}$ does too, and equal dimensions make its proper image all of $A^{\lor \lor}$ . The quotient-torsor argument makes it a finite flat isogeny.

6.1step 4.1step 5.1algebra

Degrees multiply for compositions of isogenies, and by step 4.1 applied to $φ_{L}$ we have $\deg φ_{L}^{\lor} = \deg φ_{L}$ . Taking degrees in $φ_{L} = φ_{L}^{\lor} \circ κ_{A}$ gives $\deg φ_{L} = \deg φ_{L} \cdot \deg κ_{A}$ , hence $\deg κ_{A} = 1$ ; a finite flat morphism of degree one is an isomorphism, so $κ_{A} : A \to A^{\lor \lor}$ is an isomorphism.

7.1F1F2F6step 1.2step 1.3algebra∎

Finally, $(id)^{\lor} = id$ and $(g \circ f)^{\lor} = f^{\lor} \circ g^{\lor}$ are immediate from $(g \circ f)_{T}^{*} = f_{T}^{*} g_{T}^{*}$ . For homomorphisms $f, g : A \to C$ with common source and target, the cube identity gives $m_{C}^{*} N ≅ p_{1}^{*} N \otimes p_{2}^{*} N$ for every degree-zero rigidified bundle $N$ on $C_{T}$ [F2]; hence $(f + g)_{T}^{*} N ≅ f_{T}^{*} N \otimes g_{T}^{*} N$ , which is $f^{\lor} + g^{\lor}$ under the tensor group law of $A^{\lor}$ . The identity $f^{\lor \lor} \circ κ_{A} = κ_{C} \circ f$ follows from the definition of $κ$ by the switched normalized Poincare bundles and uniqueness of representing morphisms, since both sides are represented by the same pullback of the switched bundle.

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Theta extensions, splitting and isotropic descent

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be an abelian variety over a field $k$ and let $N$ be an invertible sheaf on $A$ , with Mumford homomorphism $φ_{N} : A \to A^{\lor}$ and $K (N) = \ker φ_{N}$ the closed subgroup scheme of Homogeneous bundles and Mumford surjectivity.

(a) [theta group] For every finite subgroup scheme $H \subseteq K (N)$ the theta group $Θ (N)_{H}$ is a central extension of fppf sheaves of groups $0 \to G_{m} \to Θ (N)_{H} \to H \to 0$ whose commutator factors through an alternating bilinear pairing $e_{N} : H \times H \to G_{m}$ . The pairing satisfies $e_{f^{*} L} = e_{L} \circ (f, f)$ on subgroups mapped by the homomorphism $f$ into $K (L)$ , and $e_{L \otimes M} = e_{L} e_{M}$ on subgroups of $K (L) \cap K (M)$ ; moreover $e_{L} = 1$ whenever $[L] \in {Pic}^{0}$ .

(b) [splitting of commutative extensions] Let $k$ be algebraically closed and let $0 \to G_{m} \to E \to H \to 0$ be an extension of commutative fppf sheaves of groups with $H$ finite commutative. Then the extension splits: there is a homomorphism $H \to E$ with composite $H \to E \to H$ equal to the identity.

(c) [isotropic descent] Let $k$ be algebraically closed, let $H \subseteq K (N)$ be finite and suppose $e_{N}$ is trivial on $H \times H$ . Then $N$ admits an $H$ -linearization and descends along the isogeny $f : A \to A / H$ : there is a line bundle $L$ on the abelian variety $A / H$ with $f^{*} L ≅ N$ .

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ over a field $k$ , an invertible sheaf $N$ on $A$ with $φ_{N} : A \to A^{\lor}$ and $K (N) = \ker φ_{N}$ , and a finite subgroup scheme $H \subseteq K (N)$ .

[F1]

The dual abelian variety and the normalized Poincare bundle exist and satisfy the all-test universal property; the Mumford map is a homomorphism with $\ker φ = {Pic}^{0}$ as a sheaf on all tests (Finite-field descent of the dual and the Poincare bundle, Homogeneous bundles and Mumford surjectivity).

[F2]

Automorphisms of an invertible sheaf are scalars: $Aut (N) = G_{m}$ as an fppf sheaf, and rigidified line bundles have no nontrivial automorphisms, so the only automorphisms of a translation isomorphism $t_{x}^{*} N \to N$ compatible with a fixed trivialization are scalars (Rigidification and effective descent of line bundles).

[F3]

Cartier duality is an exact contravariant equivalence on finite commutative $k$ -group schemes, $μ_{n}^{D} ≅ (Z / n)_{k}$ , and a finite commutative $H$ of rank $n$ is killed by $n$ (Finite Cartier duality, exactness and exponent).

[F4]

For a finite subgroup $H$ of a separated finite-type group scheme the quotient $A / H$ exists as an abelian variety and $A \to A / H$ is a faithfully flat $H$ -torsor of finite presentation; every homomorphism of finite-type $k$ -group schemes with trivial kernel is a closed immersion (Normal subgroup quotients of finite-type group schemes exist as fppf scheme quotients, Finite-type algebraic group monomorphisms are closed immersions).

[F5]

Modules and algebras with descent data along faithfully flat maps are effectively descended, and morphisms of schemes descend along fppf covers; an fppf-covering morphism is submersive, so images and open conditions may be checked after the cover (Faithfully flat descent of modules and algebras is effective, Scheme morphisms satisfy fppf descent, Fpqc covers are universally submersive).

[F6]

A nonempty finite scheme over an algebraically closed field has a rational point (Over an algebraically closed field, every maximal ideal is an evaluation ideal); for a unit $u$ on a scheme $T$ the finite free cover $T [t] / (t^{n} - u)$ is faithfully flat and makes $u$ an $n$ -th power, so $x \mapsto x^{n}$ on $G_{m}$ is an fppf epimorphism (A flat ring map is faithfully flat exactly when it detects proper ideals and is surjective on spectra).

Proof

technique · direct: build the theta group from translation isomorphisms, split commutative $\mathbf G_m$-extensions of finite groups by finite Cartier duality, and conclude descent of isotropic line bundles along the quotient

1.1F1givenconstruct

Define the theta functor $Θ (N)$ on $k$ -schemes by letting $Θ (N) (T)$ be the set of pairs $(x, α)$ with $x \in A (T)$ and $α : t_{x}^{*} N_{T} \to N_{T}$ an isomorphism of line bundles on $A_{T}$ , where $t_{x} : A_{T} \to A_{T}$ is translation. The product $(x, α) (y, β) = (x + y, α \circ t_{x}^{*} β)$ makes $Θ (N)$ a group sheaf for the fppf topology, with unit $(0, id)$ ; the projection $π : Θ (N) \to A$ , $(x, α) \mapsto x$ is a homomorphism onto $K (N)$ , as an fppf sheaf: for $x \in K (N) (T)$ the difference $t_{x}^{*} N_{T} \otimes N_{T}^{- 1}$ is pulled back from a line bundle on $T$ , which becomes trivial on an fppf cover, and conversely such a local isomorphism makes its rigidified Picard class zero.

1.2F1F2givenalgebra

The scalars act on each pair by $λ \cdot (x, α) = (x, λ \circ α)$ , exhibiting $G_{m} \subseteq Θ (N)$ as a central subgroup with $π^{- 1} (0) = G_{m}$ ; conjugation by $(x, α)$ on $G_{m}$ is trivial because $G_{m}$ is commutative, and the commutator $[(x, α), (y, β)] = (x, α) (y, β) (x, α)^{- 1} (y, β)^{- 1}$ lies in $π^{- 1} (0) = G_{m}$ since $K (N)$ is commutative. Its value is the scalar $α t_{x}^{*} β (t_{y}^{*} α)^{- 1} β^{- 1}$ (with the evident identifications), which depends only on the classes of $α, β$ modulo scalars by [F2]; thus it defines a pairing $e_{N} : K (N) \times K (N) \to G_{m}$ , alternating because $[(x, α), (x, α)] = 1$ , and bilinear because the commutator is multiplicative in each variable. The identity $e_{f^{*} L} = e_{L} \circ (f, f)$ is immediate from pullback of automorphisms, $e_{L \otimes M} = e_{L} e_{M}$ from the tensor product of automorphisms, and $e_{L} = 1$ for $[L] \in {Pic}^{0}$ since then $K (L) = A$ and the pairing $e_{L} : A \times A \to G_{m}$ is constant on the complete variety $A$ .

1.3F3F6givenconstruct

We prove (b). Let $0 \to G_{m} \to E \overset{π}{\to} H \to 0$ be a commutative extension with $H$ finite commutative of rank $n$ , and let $k$ be algebraically closed. Since $H$ is killed by $n$ [F3], the endomorphism $[n]_{E}$ lands in $G_{m} = \ker π$ , so $[n]_{E}$ is a homomorphism $E \to G_{m}$ whose restriction to $G_{m}$ is $t \mapsto t^{n}$ . Let $F = \ker ([n]_{E} : E \to G_{m})$ , a subgroup sheaf of $E$ containing $μ_{n} = G_{m} [n]$ . For an fppf-local point $h \in H$ lift to $e \in E$ ; then $[n]_{E} e = t \in G_{m}$ and $x \mapsto x^{n}$ is an fppf epimorphism on $G_{m}$ [F6], so locally $t = s^{- n}$ and $e s \in F$ maps to $h$ ; hence $F \to H$ is an fppf epimorphism with kernel $μ_{n}$ , i.e. $F$ is a $μ_{n}$ -torsor over the finite scheme $H$ and is therefore representable and finite.

2.1step 1.2givenconstruct

Restricting along the closed immersion $H \subseteq K (N)$ gives the central extension $0 \to G_{m} \to Θ (N)_{H} = π^{- 1} (H) \to H \to 0$ of fppf group sheaves whose commutator is the restriction of $e_{N}$ ; this is statement (a).

2.2F3F6step 1.3construct

By Cartier exactness [F3] the dual $F^{D} \to (μ_{n})^{D} ≅ Z / n$ is an fppf epimorphism of finite commutative group schemes. As $F$ is nonempty finite over the algebraically closed field $k$ , it has a rational point, so $F^{D} (k) \neq \emptyset$ , and surjectivity on $k$ -points (a morphism of finite type schemes over an algebraically closed field which is fppf surjective is surjective on closed points) provides a character $χ : F \to G_{m}$ whose restriction to $μ_{n}$ is the tautological character.

3.1F3F5step 2.2algebra

Consider the multiplication morphism $μ : G_{m} \times F \to E$ , $(t, f) \mapsto t f$ . It is an fppf epimorphism: for $e \in E$ with $t = [n]_{E} e$ and locally $t = s^{n}$ , the element $s^{- 1} e$ lies in $F$ , so $e = s (s^{- 1} e)$ ; its kernel is ${(t, f) : t f = 1} = {(s, s^{- 1}) : s \in μ_{n}}$ , so $μ$ is a $μ_{n}$ -torsor. The morphism $ψ (t, f) = t χ (f)$ satisfies $ψ (t ζ, ζ^{- 1} f) = t ζ χ (ζ)^{- 1} χ (f) = t χ (f)$ for $ζ \in μ_{n}$ , hence is invariant under the kernel and descends along the fppf cover $μ$ to a morphism $r : E \to G_{m}$ [F5]. On $G_{m} \subset E$ one has $r (t) = t χ (1) = t$ , and $r$ is a homomorphism because $ψ$ is multiplicative and $E$ , hence $F$ , is commutative. Therefore $E \to G_{m} \times \ker r$ , $e \mapsto (r (e), e r (e)^{- 1})$ , is an isomorphism with inverse $(t, u) \mapsto t u$ , and $\ker r \to H$ is an isomorphism; the extension splits.

4.1step 2.1step 3.1givenconstruct

We prove (c). If $e_{N}$ is trivial on $H \times H$ , then every commutator in $Θ (N)_{H}$ is trivial, so $Θ (N)_{H}$ is commutative; by (b) the extension $0 \to G_{m} \to Θ (N)_{H} \to H \to 0$ splits. A homomorphic section $σ : H \to Θ (N)_{H}$ assigns to each $h \in H (T)$ an isomorphism $α_{h} : t_{h}^{*} N_{T} \to N_{T}$ with $α_{h + h^{'}} = α_{h} \circ t_{h}^{*} α_{h^{'}}$ , which is precisely an $H$ -linearization of $N$ covering the translation action of $H$ on $A$ .

5.1F4F5step 4.1algebra∎

With this linearization, $N$ is an $H$ -equivariant line bundle on the $H$ -torsor $f : A \to A / H$ [F4]; the fppf descent equivalence of modules [F5] produces a line bundle $L$ on $A / H$ with $f^{*} L ≅ N$ . The descended module is invertible of rank one because $A \to A / H$ is faithfully flat and rank-one local freeness is checked after such base change; this proves (c).

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Poincare cohomology at the identity

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be an abelian variety of dimension $g$ over a field $k$ , let $A^{\lor}$ be its dual and let $P$ be the normalized Poincare bundle on $A \times_{k} A^{\lor}$ , with second projection $p_{2} : A \times_{k} A^{\lor} \to A^{\lor}$ . Then $R^{i} p_{2, *} P = 0$ for $i \neq g$ , and $R^{g} p_{2, *} P ≅ k (0)$ is the length-one skyscraper sheaf at the origin of $A^{\lor}$ . In particular the statement retains infinitesimal scheme lengths and does not replace the origin by its reduced point.

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ of dimension $g$ over a field $k$ , its dual $A^{\lor}$ with normalized Poincare bundle $P$ on $A \times_{k} A^{\lor}$ and second projection $p_{2}$ .

[F1]

The Poincare bundle is the universal rigidified line bundle; for every $b \in A^{\lor}$ the fibre $P_{b} = P ∣_{A \times {b}}$ is the corresponding degree-zero line bundle on $A$ , and it is trivial exactly at $b = 0$ (Finite-field descent of the dual and the Poincare bundle, Homogeneous bundles and Mumford surjectivity).

[F2]

Every nontrivial homogeneous invertible sheaf on $A$ has vanishing cohomology in all degrees, and $\ker φ = {Pic}^{0}$ for all tests (Homogeneous bundles and Mumford surjectivity).

[F3]

For a proper finite presentation flat family $f : X \to Spec R$ there are an integer $r \geq 0$ and a bounded complex $K$ of finite free $R$ -modules concentrated in degrees $0, \dots, r$ , together with a canonical isomorphism $H^{q} (K \otimes_{R} A^{'}) ≅ H^{q} (X_{A^{'}}, F_{A^{'}})$ for every $R$ -algebra $A^{'}$ ; the sheaves $R^{i} p_{2, *} P$ are coherent, and a coherent module on $Spec R$ supported at the closed point has finite length (Universal finite projective cohomology complex over any base, Cohomology and base change for proper flat coherent families).

[F4]

Cohomology of coherent sheaves on an abelian variety is computed by Cech complexes with Kunneth, Leray and cup-product structure, and vanishes above $g = \dim A$ ; $H^{0} (A, O_{A}) = k$ and $H^{g} (A, O_{A}) = k$ by Serre duality and triviality of the canonical bundle (Cech cohomology computes quasi-coherent cohomology on a separated scheme, Kunneth over a field, Cup-product laws, Leray spectral sequence for sheaf cohomology, Grothendieck vanishing on a Noetherian space, Serre duality for locally free sheaves on a smooth projective variety).

[F5]

Over a regular local ring $R$ of dimension $g$ with regular parameter system $x_{1}, \dots, x_{g}$ , the powers $x_{1}^{n}, \dots, x_{g}^{n}$ form a regular sequence, so each Koszul complex $K (x_{1}^{n}, \dots, x_{g}^{n})$ has no cohomology below degree $g$ ; filtered colimits of modules are exact, so the Cech complex $E$ on $x_{1}, \dots, x_{g}$ has $H^{i} (E) = 0$ for $i < g$ (Regular Sequences Give Acyclic Koszul Complexes, regular local rings are domains and cohen macaulay).

[F6]

Minimal finite free complexes over a local ring are obtained by splitting off contractible summands; their differentials vanish after reduction to the residue field, and a finite-length submodule of a free module over a domain of positive dimension is zero (Assuming the Axiom of Choice, Nakayama's lemma, Completion of a finite module is extension of scalars).

[F7]

Two projective resolutions of the same module over a ring are homotopy equivalent, and chain maps between resolutions lift the identity; enough projectives and comparison lifts are available (Projective resolutions of the same object are homotopy equivalent over that object, A morphism has a comparison lift between the supplied projective resolutions, A chosen chain of projective epimorphisms gives a projective resolution).

Proof

technique · direct: reduce to the local ring at the origin, replace $Rp_{2,*}\mathcal P$ by a finite free universal cohomology complex, prove vanishing below degree $g$ by a Cech/Koszul finite-length argument, and identify the terminal cokernel with $k$ by the universal property and comparison with the Koszul resolution

1.1F1F2F4givenconstruct

Put $B = A^{\lor}$ and $R = O_{B, 0}$ , a regular local ring of dimension $g$ . For $b \neq 0$ the fibre $P_{b}$ is a nontrivial degree-zero bundle on $A$ , hence a nontrivial homogeneous bundle; by [F2] all its cohomology vanishes, so the formation of $R^{i} p_{2, *} P$ is supported at the closed point $0 \in B$ , and by [F4] $R^{i} p_{2, *} P = 0$ for $i < 0$ and $i > g$ .

1.2F5F6givenalgebra

Local acyclicity claim. Let $C$ be a bounded complex of finite free modules over the $g$ -dimensional regular local ring $R$ with $C^{i} = 0$ for $i < 0$ and all $H^{i} (C)$ of finite length. Then $H^{i} (C) = 0$ for $i < g$ . For the proof choose a regular parameter system $x_{1}, \dots, x_{g}$ and let $E$ be the augmented Cech complex $R \to ⨁_{i} R [x_{i}^{- 1}] \to \dots \to R [(x_{1} \dots x_{g})^{- 1}]$ in degrees $0, \dots, g$ . By [F5] $E$ is the filtered colimit of the Koszul complexes on $x_{1}^{n}, \dots, x_{g}^{n}$ , each with no cohomology below $g$ , so $H^{i} (E) = 0$ for $i < g$ . Form the bounded double complex $C \otimes_{R} E$ . Computing $E$ first, its $E_{1}$ page vanishes in degrees below $g$ because $E$ does and $C$ starts in degree $0$ , so the total cohomology vanishes below $g$ . Computing $C$ first, each $E^{q}$ is a direct sum of localizations and hence flat, so $E_{2}^{p, q} = H^{p} (C) \otimes_{R} E^{q}$ ; since $H^{p} (C)$ is finite length it is killed by a power of the maximal ideal, and $x_{i}$ lies in the maximal ideal, so every nonempty localization of $H^{p} (C)$ vanishes: $E_{2}^{p, q} = 0$ for $q \geq 1$ and $E_{2}^{p, 0} = H^{p} (C)$ . The second spectral sequence degenerates and identifies the total cohomology in degree $p$ with $H^{p} (C)$ ; comparing with the first computation gives $H^{p} (C) = 0$ for $p < g$ .

2.1F3F4step 1.1construct

Since $A \times B \to B$ is proper and $P$ is flat over $B$ , applying [F3] over $Spec R$ produces a bounded finite free complex; cancel its contractible summands over the local ring $R$ to obtain a minimal complex $K$ . Its reduction has zero differentials, and fibre cohomology vanishes above $g$ , so its terms vanish above $g$ . Thus $K$ lies in degrees $0, \dots, g$ with canonical isomorphisms $H^{i} (K \otimes_{R} A^{'}) ≅ H^{i} (A_{A^{'}}, P_{A^{'}})$ for every $R$ -algebra $A^{'}$ ; in particular $H^{i} (K) ≅ (R^{i} p_{2, *} P)_{0}$ is a finite-length $R$ -module by [F3] and step 1.1, and $H^{i} (K \otimes_{R} k) = H^{i} (A, O_{A})$ , with degree-zero and degree- $g$ dimensions one by [F4]; after minimalization these determine the endpoint ranks.

3.1step 2.1step 1.2construct

Apply the local acyclicity claim to $K$ : $H^{i} (K) = 0$ for $i < g$ , so by step 1.1 the complex $K$ has cohomology only in degree $g$ , where $H^{g} (K)$ is a finite-length $R$ -module. Form the shifted dual complex $C = {Hom}_{R} (K, R) [- g]$ , so $C^{i} = {Hom}_{R} (K^{g - i}, R)$ with the usual dual signs; it is again a bounded complex of finite free modules in degrees $0, \dots, g$ . Away from the closed point the finite-length cohomology of $K$ localizes to zero, so $K$ becomes split exact there and hence so does its dual $C$ ; therefore each $H^{i} (C)$ is supported at the closed point and has finite length. The local acyclicity claim applied to $C$ gives $H^{i} (C) = 0$ for $i < g$ .

4.1F4F6step 2.1step 3.1algebra

If $g = 0$ , the smooth geometrically connected zero-dimensional pointed variety $A$ is $Spec k$ , as is its dual, so the conclusion is immediate. Suppose $g > 0$ . The minimal complex $K$ of step 2.1 has endpoint ranks one, since $H^{0} (A, O_{A}) = H^{g} (A, O_{A}) = k$ . Thus $K^{0} ≅ K^{g} ≅ R$ , and the dual complex $C$ has endpoint ranks one too. By step 3.1 its sole cohomology is the terminal cokernel $H^{g} (C) = R / J$ , where $J$ is the ideal generated by the entries of its last differential, equivalently by the entries of the first differential $d_{K}^{0} : R \to K^{1}$ . The module $R / J$ has finite length, and minimality ensures $J \subseteq m_{R}$ .

5.1F1F4F6step 4.1construct

Tensor $K$ with $R / J$ . Its first differential vanishes because all entries lie in $J$ , and it has no negative terms, so $H^{0} (K \otimes_{R} R / J) = R / J$ . The universal cohomology identification of [F3] turns the basis vector of $K^{0}$ into a section of $P$ over $A \times_{k} Spec R / J$ . Its reduction is a nonzero section of $O_{A}$ , hence nowhere zero. Since the Artinian local scheme $Spec R / J$ has the same underlying point as $Spec k$ , Nakayama makes this section a trivialization everywhere; normalize its value at the identity to give a rigidified trivialization. The universal property [F1] therefore makes the classifying map $Spec R / J \to B$ factor through the origin. As this map is the natural map induced by $R = O_{B, 0}$ , it follows that $m_{R} \subseteq J$ . Hence $J = m_{R}$ and $H^{g} (C) ≅ k$ .

6.1F7step 3.1step 5.1algebra

Thus $C$ is a minimal finite free complex over $R$ concentrated in degrees $0, \dots, g$ with $H^{i} (C) = 0$ for $i < g$ and $H^{g} (C) ≅ k$ ; it is a minimal free resolution of $k$ shifted by $g$ . Comparing it with the Koszul resolution of $k$ on a regular parameter system by chain lifting [F7] shows that the two complexes are homotopy equivalent, and dualizing back by ${Hom}_{R} (-, R) [- g]$ (an exact anti-equivalence on finite free complexes, carrying the Koszul resolution to its dual) yields that $K$ is quasi-isomorphic to $k [- g]$ : the dual Koszul complex has sole cohomology $k$ in degree $g$ . Therefore $H^{i} (K) = 0$ for $i \neq g$ and $H^{g} (K) ≅ k$ as an $R$ -module.

7.1step 1.1step 2.1step 6.1algebra∎

Translating step 6.1 back through the identification $H^{i} (K) = (R^{i} p_{2, *} P)_{0}$ of step 2.1 gives $R^{i} p_{2, *} P = 0$ for $i \neq g$ and $R^{g} p_{2, *} P$ a coherent sheaf supported at $0$ with stalk $k$ , i.e. the length-one skyscraper $k (0)$ ; the identification is natural in the local ring, so the global statement follows.

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Finite translate completion and uniqueness

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ , let $R^{s h}$ be a strict henselization, and let $X$ be a smooth separated faithfully flat finite-type $R^{s h}$ -scheme with a strict $R^{s h}$ -birational group law. Then finitely many section translates of $X$ yield a smooth separated finite-type $R^{s h}$ -scheme $Y$ on which the multiplication of $X \times X$ is everywhere defined, and $Y$ is an $R^{s h}$ -group scheme containing $X$ as a fibre-dense open subscheme. Any two group completions of the birational law are canonically isomorphic. The uniqueness assertion also holds after arbitrary base change: more generally it holds for smooth separated finitely presented group schemes over a base $S$ containing the same smooth faithfully flat finitely presented fibre-dense open $X / S$ with the same strict law.

Facts & Assumptions

Given: AC and DC, a DVR $R$ with fraction field $K$ and residue field $k$ , a strict henselization $R^{s h}$ , and a smooth separated faithfully flat finite-type $R^{s h}$ -scheme $X$ with a strict birational group law $m$ .

[F1]

The strict graph has open-immersion two-coordinate projections and both-projection dense images; section translation is defined at a point exactly when the law is defined at the pair (Strict law graph calculus). Gluing a left translate along its closed graph preserves smoothness, separatedness, finite type and strictness (Separated translate gluing).

[F2]

Over the strictly henselian base every dense open of either fibre is met by a section, and sections meet all fibre-dense opens (Strict henselization of a DVR and smooth sections); rational maps descend along faithfully flat maps and agree on schematically dense opens of separated targets (S-dense open subschemes and S-rational maps, An S-rational map defined after a faithfully flat smooth base change is defined, Scheme morphisms satisfy fppf descent, Agreement on a schematically dense open).

Proof

technique · direct: enlarge by translates until the multiplication domains stabilize, then descend and verify the group axioms

1.1F1givenconstruct

Successively glue translates $X^{(i + 1)} = X^{(i)} \cup X^{(i)} (a_{i + 1})$ whenever a section translation of the original $X$ is not defined everywhere as a map $X ⇢ X^{(i)}$ . Let $Q_{i} \subseteq X \times X$ be the domain of its original multiplication with values in $X^{(i)}$ . By the graph open-immersion property [F1] these are increasing opens of the fixed Noetherian scheme $X \times X$ . If the next translation is undefined at $b$ , its newly glued translate defines it there; graph calculus then puts $(a_{i + 1}, b)$ in $Q_{i + 1} ∖ Q_{i}$ . Such strict increases cannot continue indefinitely in a Noetherian space. Thus for a finite enlargement $Y$ , every original section translation $t_{a} : X \to Y$ is everywhere defined.

2.1F1F2step 1.1algebra

Fix a geometric pair $(x, y)$ in $X \times X$ . Strict graph calculus [F1] makes the auxiliary locus of $w$ where $w^{- 1} x \in X$ and $(w^{- 1} x, y)$ is in the original law domain open and fibre-dense in $X$ . By [F2] some section $a$ meets this locus: on the special fibre use smooth section lifting, and on the generic fibre use the generic-open assertion on a component with nonempty special fibre. Then the map $(x, y) \mapsto a ((a^{- 1} x) y)$ is defined near the pair, since its inner product lies in the original $X$ and $t_{a} : X \to Y$ is everywhere defined by step 1.1. Associativity identifies this map with the original rational multiplication. It follows that the extended strict law on $Y$ is a morphism on the entire original $X \times X$ .

3.1F1F2step 2.1algebra

To extend the law to all of $Y \times Y$ , choose an auxiliary $a \in X$ at a generic point of the fibre over a given pair $(b, c) \in Y \times Y$ . The strict graph projections [F1] put $b a \in X$ and $a^{- 1} c \in X$ on a fibre-dense open auxiliary locus. Their product is defined by step 2.1, and associativity gives $b c = (b a) (a^{- 1} c)$ . This gives the rational law a regular representative on a smooth faithfully flat auxiliary cover, so domain descent [F2] makes it regular at $(b, c)$ . The same argument makes the division map $(b, c) \mapsto b^{- 1} c$ regular everywhere. The strict translation monomorphism into the relative birational-map group, supplied by [F1], now identifies $Y (T)$ with a subset closed under multiplication and division for every test $T$ . It is nonempty since faithful flatness and smoothness over the strictly henselian DVR give an $R^{s h}$ -section. Hence it is a subgroup: division gives the unit and inverses, and associativity holds by the strict law and schematic density. These natural operations are scheme morphisms by their construction, so $Y$ is the claimed smooth separated finite-type group completion.

4.1F1F2step 3.1algebra∎

For uniqueness over a base $S$ , let $Y_{1}, Y_{2}$ be smooth separated finitely presented group completions containing the same smooth faithfully flat finitely presented fibre-dense $X$ . Each $Y_{i}$ acts faithfully by birational translations on $X_{T}$ for every test $T$ : the domain $X_{T} \cap y^{- 1} X_{T}$ is fibre-dense, and equality of translations implies equality of the translating sections after the faithfully flat dense domain cover of $T$ . The multiplication map $q_{i} : X \times_{S} X \to Y_{i}$ is smooth, since it is a restriction of group multiplication, and surjective: in each geometric fibre, for a prescribed $y$ , the dense opens $X$ and $y X^{- 1}$ intersect. It is quasi-compact and locally finitely presented. On the kernel pair of $q_{1}$ , two pairs from $X^{2}$ have the same product in $Y_{1}$ , hence the same product of birational translations of $X$ by [F1]; faithfulness for $Y_{2}$ makes their $q_{2}$ -images equal. Fppf morphism descent [F2] gives $Y_{1} \to Y_{2}$ with composite $q_{1}$ equal to $q_{2}$ . Reversing the roles gives an inverse, by surjectivity of the covers. The isomorphism fixes $X$ (compare translations, using their faithful action) and preserves group multiplication by the same comparison. Faithfulness also proves uniqueness. This proof applies to arbitrary pullback bases, including the tensor-product bases used for descent.

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Symmetric homomorphisms are Mumford maps

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be an abelian variety over a field $k$ , and let $λ : A \to A^{\lor}$ be a symmetric homomorphism, that is, $λ = λ^{\lor} \circ κ_{A}$ under the canonical biduality identification (Dual isogenies, Cartier-dual kernels and canonical biduality, Polarizations and the Mumford isogeny attached to an ample line bundle). Then there is a finite separable field extension $k^{'} \supset k$ and an invertible sheaf $L$ on $A_{k^{'}}$ with $λ_{k^{'}} = φ_{L}$ . In particular the conclusion holds over every separably closed field, including fields of characteristic two.

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ over a field $k$ and a symmetric homomorphism $λ : A \to A^{\lor}$ .

[F1]

Dual isogenies, their Cartier-dual kernels, degrees and the canonical biduality $κ_{A}$ are constructed in Dual isogenies, Cartier-dual kernels and canonical biduality; the Poincare bundle is the universal normalized bundle (Finite-field descent of the dual and the Poincare bundle).

[F2]

For $λ : A \to A^{\lor}$ put $M = (id, λ)^{*} P$ ; the biextension identities give $φ_{M} = λ + λ^{\lor} \circ κ_{A}$ , so for symmetric $λ$ one has $φ_{M} = 2 λ$ , and the commutator pairing $e_{N}$ of $N = M^{2}$ has values in $μ_{4}$ on $A [4]$ (Dual isogenies, Cartier-dual kernels and canonical biduality, Theta extensions, splitting and isotropic descent).

[F3]

If $k$ is algebraically closed and $H \subseteq K (N)$ is finite with $e_{N}$ trivial on $H \times H$ , then $N$ descends along $A \to A / H$ : there is a line bundle $L$ with $N ≅ [2]^{*} L$ when $H = A [2]$ (Theta extensions, splitting and isotropic descent).

[F4]

Over an algebraically closed field the rigidified relative Picard functor is represented by a separated locally finite-type group scheme with a universal rigidified bundle; the Mumford map depends only on the class of the bundle modulo ${Pic}^{0}$ , and $\ker φ = {Pic}^{0}$ as a sheaf (Picard representation by generic quotient and translates, Homogeneous bundles and Mumford surjectivity).

[F5]

The locus where coherent equations vanish is represented by closed subschemes compatible with base change, projective twists of coherent ideals on a projective family are eventually globally generated, their cohomology is finite and vanishes in high degree, and finite-field scheme descent is effective when the finite descent orbits lie in affine opens (Finite field descent is effective for schemes with affine-contained descent orbits, Eventual generation of coherent projective twists, Projective coherent finiteness and large twist vanishing, Cohomology and base change for proper flat coherent families, Scheme morphisms satisfy fppf descent).

[F6]

Multiplication $[n] : A \to A$ is finite faithfully flat of rank $∣ n ∣^{2 g}$ and $[2] : A [4] \to A [2]$ is an fppf epimorphism (Nonzero multiplication on an abelian variety is finite and faithfully flat); a nonempty smooth finite-type scheme over a field has a closed point with finite separable residue field (A nonempty smooth scheme has a finite separable point).

Proof

technique · direct: realize a symmetric homomorphism over an algebraic closure by descending through the isotropic subgroup $A[2]$, then represent the locus of realizing bundles as a smooth torsor and apply the finite-separable-point theorem

1.1F2givenalgebra

Work first over an algebraic closure $\overset{ˉ}{k}$ of $k$ and write $λ$ again for $λ_{\overset{ˉ}{k}}$ . Put $M = (id, λ)^{*} P$ and $N = M^{2}$ . The Poincare biextension identities expand $P (x + y, λ (x + y))$ into $P (x, λ x) \otimes P (y, λ y) \otimes P (x, λ y) \otimes P (y, λ x)$ . Thus the normalized square family of $M$ is the product of the two cross families. The first represents $λ$ , and the switched second represents $λ^{\lor} \circ κ_{A}$ by the defining dual-pullback and biduality identities in [F2]. Hence $φ_{M} = λ + λ^{\lor} \circ κ_{A} = 2 λ$ . Tensor multiplicativity gives $φ_{N} = 4 λ$ , so $A [4] \subseteq K (N)$ .

1.2F4givenconstruct

Now let $k$ be arbitrary and consider the fppf sheaf $Z$ on $k$ -schemes whose $T$ -points are the relative Picard classes of invertible sheaves $L$ on $A_{T}$ , normalized along the identity with $φ_{L} = λ_{T}$ . If $Z (T) \neq \emptyset$ and $L \in Z (T)$ , then $L^{'} \mapsto L^{'} \otimes L^{- 1}$ identifies $Z (T)$ with $\ker (φ) (T) = {Pic}^{0} (A_{T}) = A^{\lor} (T)$ by [F4], so $Z$ is an $A^{\lor}$ -torsor for the fppf topology.

2.1F6F2givenalgebra

Restrict the alternating bilinear commutator pairing $e_{N}$ to $A [4] \times A [4]$ , which is legitimate because $A [4] \subseteq K (N)$ by step 1.1. Its values lie in $μ_{4}$ : bilinearity gives $e_{N} (x^{'}, y^{'})^{4} = e_{N} (4 x^{'}, y^{'}) = 1$ . For $x, y \in A [2]$ , lift fppf-locally to $x^{'}, y^{'} \in A [4]$ with $x = 2 x^{'}$ and $y = 2 y^{'}$ by [F6]. Then $e_{N} (x, y) = e_{N} (x^{'}, y^{'})^{4} = 1$ . Descent of equality proves isotropy on the full group scheme $A [2] \times A [2]$ , including characteristic two. This uses $e_{N}$ on its actual domain and does not extend $e_{M}$ outside $K (M)$ .

3.1F3F2F6step 1.1step 2.1algebra

Apply the isotropic descent of [F3] with $H = A [2]$ and the quotient morphism $[2] : A \to A$ : since $e_{N}$ is trivial on $A [2] \times A [2]$ , the line bundle $N$ descends through $[2]$ , so there is an invertible sheaf $L$ on $A$ with $N ≅ [2]^{*} L$ . Then $4 λ = φ_{N} = [2]^{*} φ_{L} = 4 φ_{L}$ , using $φ_{[2]^{*} L} = [2]^{*} \circ φ_{L} \circ [2] = 4 φ_{L}$ . Therefore $λ - φ_{L}$ is a homomorphism whose image lies in $\ker [4] = A^{\lor} [4]$ , a finite group scheme; since $A$ is proper and geometrically integral, every map from $A$ to a finite affine scheme factors through $Γ (A, O_{A}) = k$ , so a pointed homomorphism to that finite scheme is zero, so $λ = φ_{L}$ .

4.1F4F5step 1.2construct

The realization over $\overset{ˉ}{k}$ in step 3.1 descends to a finite extension $K / k$ : an invertible sheaf is described by finitely many generators and transition functions on a finite affine cover, and the equality of its Mumford morphism with $λ$ is described by finitely many equations on affine covers; all their algebraic coefficients lie in a finite extension. Call this realizing bundle $L_{0}$ on $A_{K}$ . Tensoring $L_{0}$ with the universal Poincare bundle identifies $Z_{K}$ with $A_{K}^{\lor}$ on all tests by the kernel equality [F4]. In particular $Z$ is fppf-locally nonempty, as required in step 1.2, and $Z_{K}$ carries the canonical descent datum obtained from equality of its classifying functor over $K \otimes_{k} K$ . This datum satisfies the cocycle by uniqueness of the functor identification. The scheme $Z_{K}$ is projective, since it is an abelian variety. Its finite descent orbits lie in affine opens: choose closed specializations of the finitely many orbit points and a sufficiently high very ample power; Serre vanishing and eventual generation in [F5] supply a section nonzero at each specialization, whose nonvanishing affine open contains the whole orbit. The finite-field descent theorem [F5] therefore descends $Z_{K}$ to a finite-type $k$ -scheme representing $Z$ , including inseparable $K / k$ . This argument requires only the represented dual and its universal bundle and does not presume that the full Picard scheme has already been constructed over $k$ .

5.1F1F6step 3.1step 1.2step 4.1algebra∎

The scheme $Z$ is smooth over $k$ : it is an $A^{\lor}$ -torsor by step 1.2, $A^{\lor}$ is smooth over $k$ by [F1], and smoothness is fppf-local. It is nonempty because after extending scalars to $\overset{ˉ}{k}$ the realization $λ = φ_{L}$ of step 3.1 gives a $\overset{ˉ}{k}$ -point of $Z$ . Since $Z$ is a nonempty smooth finite-type $k$ -scheme, the finite-separable-point theorem [F6] provides a closed point $z \in Z$ whose residue field $k^{'}$ is finite and separable over $k$ ; the tautological bundle at $z$ is an invertible sheaf $L$ on $A_{k^{'}}$ with $φ_{L} = λ_{k^{'}}$ , as required.

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The square degree of a Mumford map

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be an abelian variety of dimension $g$ over a field $k$ and let $L$ be a nondegenerate invertible sheaf on $A$ , that is, $K (L) = \ker φ_{L}$ is finite. Then $\deg φ_{L} = χ (A, L)^{2},$ where $\deg φ_{L}$ is the finite locally free rank of the isogeny $φ_{L}$ ; the identity retains characteristic-dividing degrees and the full nonreduced scheme length of $K (L)$ . Consequently the degree of every polarization of $A$ (Polarizations and the Mumford isogeny attached to an ample line bundle) is a perfect square.

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ of dimension $g$ over a field $k$ , a nondegenerate invertible sheaf $L$ on $A$ , and the Mumford map $φ_{L} : A \to A^{\lor}$ .

[F1]

For the normalized Poincare bundle $P$ on $A \times_{k} A^{\lor}$ one has $R^{i} p_{2, *} P = 0$ for $i \neq g$ and $R^{g} p_{2, *} P = k (0)$ , the length-one skyscraper at the origin (Poincare cohomology at the identity).

[F2]

The Mumford map is a homomorphism compatible with field extension, and $(id \times φ_{L})^{*} P ≅ Λ (L) = m^{*} L \otimes p_{1}^{*} L^{- 1} \otimes p_{2}^{*} L^{- 1} \otimes π^{*} e^{*} L$ on $A \times_{k} A$ (Polarizations and the Mumford isogeny attached to an ample line bundle, Homogeneous bundles and Mumford surjectivity). The dual has dimension $g$ and is geometrically integral (Finite-field descent of the dual and the Poincare bundle, Abelian varieties over a field). Finite flatness for the nondegenerate map is proved in step 1.1, rather than assumed from a theorem whose input already is an isogeny.

[F6]

A morphism from a proper scheme to a separated scheme is proper, and proper quasi-finite morphisms are finite (Morphisms from a proper scheme to a separated one are proper, A proper quasi-finite morphism is finite). Over an algebraically closed field, the image-plus-generic-fibre dimension formula applies to irreducible classical varieties (Image dimension and the generic fibre formula). The quotient by a closed normal subgroup is represented, with faithfully flat finite-presentation projection, and a finite-type group homomorphism with trivial scheme-theoretic kernel is a closed immersion (Normal subgroup quotients of finite-type group schemes exist as fppf scheme quotients, Finite-type algebraic group monomorphisms are closed immersions). A finite flat module over a Noetherian local ring is free (A finite flat module over a local ring is free).

[F3]

Cohomology of coherent sheaves on $A$ and on $A \times_{k} A$ is computed by Cech complexes, satisfies Kunneth over a field, vanishes above the dimension, and carries the Leray spectral sequence; Euler characteristics are additive in short exact sequences and multiplicative for external tensor products (Cech cohomology computes quasi-coherent cohomology on a separated scheme, Kunneth over a field, Grothendieck vanishing on a Noetherian space, Leray spectral sequence for sheaf cohomology, Euler characteristic is additive in short exact sequences, Cup-product laws).

[F4]

Flat base change for the proper flat family $A \times A^{\lor} \to A^{\lor}$ is supplied by the exact nonnegative universal cohomology complex: tensoring its kernel and cokernel descriptions by a flat base-change ring commutes with cohomology, and the natural comparisons are the geometric base-change maps (Universal finite projective cohomology complex over any base, Cohomology and base change for proper flat coherent families).

[F5]

The abelian variety $A$ is projective by Every abelian variety over a field is projective. Translation trivializes its cotangent bundle: a basis at the identity extends by translation to a basis everywhere, and taking its top exterior power trivializes the canonical bundle (Differentials of a smooth morphism). Serre duality on this smooth projective variety gives $χ (A, L^{- 1}) = (- 1)^{g} χ (A, L)$ and $h^{i} (A, L) = h^{g - i} (A, L^{- 1} \otimes ω_{A})$ ; the canonical bundle of $A$ is trivial (Serre duality for locally free sheaves on a smooth projective variety, Abelian varieties over a field).

Proof

technique · direct: pull back the Poincare cohomology theorem along the finite flat isogeny $\varphi_{\mathcal L}$, compute $\chi(\Lambda(\mathcal L))$ twice, and deduce the square-degree formula and its invariance under field extension

1.1F2F6givenalgebra

Put $H = K (L)$ , finite by hypothesis. The morphism $φ = φ_{L} : A \to A^{\lor}$ is proper by [F6]. After any algebraically closed field extension, each nonempty fibre is a translate of $H$ as a scheme, hence finite. Thus $φ$ is quasi-finite and therefore finite by [F6]. Its geometric image is closed and irreducible. The image-dimension formula, with zero-dimensional generic fibre, gives image dimension $g$ ; since the geometrically integral target has dimension $g$ , this closed image is the whole target. Surjectivity descends to $k$ . Form the fppf quotient $q : A \to Q = A / H$ of [F6]. The induced homomorphism $u : Q \to A^{\lor}$ has trivial scheme-theoretic kernel: any kernel section lifts fppf-locally to a section $a$ of $A$ , and $φ (a) = 0$ means $a \in H$ , whose quotient class is zero. Hence $u$ is a closed immersion by [F6]; it is surjective because $φ = u q$ is surjective. The target $A^{\lor}$ is reduced, so the ideal of this surjective closed immersion is zero and $u$ is an isomorphism. Consequently $φ = q$ is faithfully flat. Together with its already proved finiteness, this makes $φ_{*} O_{A}$ a finite flat module on each Noetherian affine target chart. It is free at each local ring by [F6]; a local basis and its inverse spread to a neighbourhood because the module is finitely presented. Thus $φ$ is finite locally free. Its rank is constant on the connected target and its fibre at zero is $H$ , so that rank is $\dim_{k} O (H) = \deg φ$ . This proves the exact flatness needed for the following base change, including nonreduced $H$ .

2.1F1F2F4step 1.1construct

Consider the Cartesian square with $φ = φ_{L}$ , the morphism $(id \times φ) : A \times_{k} A \to A \times_{k} A^{\lor}$ , and projections $p_{2} : A \times_{k} A \to A$ and $p_{2} : A \times_{k} A^{\lor} \to A^{\lor}$ . By flat base change [F4] applied to [F1] we have $R p_{2, *} ((id \times φ)^{*} P) ≅ φ^{*} (R p_{2, *} P) ≅ φ^{*} (k (0) [- g]) ≅ O_{K (L)} [- g]$ , where $φ^{- 1} (0) = K (L)$ is finite of length $\deg φ$ by [F2]. Since $(id \times φ)^{*} P ≅ Λ (L)$ by [F2], the direct images of $Λ (L)$ under $p_{2}$ vanish except in degree $g$ , where they equal $O_{K (L)}$ .

3.1F2F3step 2.1algebra

The Leray spectral sequence of $p_{2}$ for $Λ (L)$ has only the $q = g$ row, supported on the finite scheme $K (L)$ ; hence $H^{n} (A \times_{k} A, Λ (L)) ≅ H^{n - g} (K (L), O_{K (L)})$ , which is $k^{\deg φ}$ for $n = g$ and zero otherwise. Therefore $χ (A \times_{k} A, Λ (L)) = (- 1)^{g} \deg φ_{L}$ , with the full scheme length, including any characteristic-dividing part.

3.2F2F3step 2.1algebra

We compute the same Euler characteristic by an automorphism. Let $σ = (m, p_{1}) : A \times_{k} A \to A \times_{k} A$ be the automorphism $(x, y) \mapsto (x + y, x)$ , with inverse $(x, y) \mapsto (y, x - y)$ . Then $Λ (L) \otimes p_{2}^{*} L = σ^{*} (L ⊠ L^{- 1}) \otimes π^{*} e^{*} L$ , where $L ⊠ L^{- 1} = p_{1}^{*} L \otimes p_{2}^{*} L^{- 1}$ and $π^{*} e^{*} L$ is the constant pullback of a line bundle from the base field; this is immediate from $σ^{*} (p_{1}^{*} L \otimes p_{2}^{*} L^{- 1}) = m^{*} L \otimes p_{1}^{*} L^{- 1}$ . Because $R p_{2, *} Λ (L) = O_{K (L)} [- g]$ is supported on the finite scheme $K (L)$ [step 2.1], tensoring by the base-pulled line bundle $p_{2}^{*} L$ does not change the Euler characteristic: it tensors the finite-length cohomology by the rank-one bundle $L ∣_{K (L)}$ , preserving all lengths. Hence $χ (Λ (L) \otimes p_{2}^{*} L) = χ (Λ (L))$ .

4.1F3F5step 3.1step 3.2algebra

Since $σ$ is an automorphism and $π^{*} e^{*} L$ is pulled back from the base, $χ (σ^{*} (L ⊠ L^{- 1}) \otimes π^{*} e^{*} L) = χ (L ⊠ L^{- 1})$ , and by Kunneth multiplicativity for external tensor products [F3] this is $χ (A, L) χ (A, L^{- 1}) = (- 1)^{g} χ (A, L)^{2}$ , the last equality by Serre duality [F5]. Combining with steps 3.1 and 3.2 gives $(- 1)^{g} \deg φ_{L} = (- 1)^{g} χ (A, L)^{2}$ , hence $\deg φ_{L} = χ (A, L)^{2}$ .

5.1F2F4step 4.1algebra∎

Finally let $λ : A \to A^{\lor}$ be a polarization. By definition $λ_{\overset{ˉ}{k}} = φ_{L}$ for an ample invertible sheaf $L$ on $A_{\overset{ˉ}{k}}$ , and $φ_{L}$ is an isogeny with $\deg φ_{L} = χ (A_{\overset{ˉ}{k}}, L)^{2}$ by step 4.1. Degree and Euler characteristic are unchanged by field extension (the degree is the rank of a finite locally free morphism, and coherent cohomology is compatible with flat field base change [F4]), so $\deg λ = χ (A_{\overset{ˉ}{k}}, L)^{2}$ is a perfect square in $Z$ .

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Effective ample-pair and group descent from a strict henselization

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ , and let $R^{s h}$ be a chosen strict henselization. Let $U$ be a smooth separated finite-type $R$ -scheme with a strict $R$ -birational group law and let $U_{R^{s h}}$ be its base change.

(a) [ample pair descent] If $(X^{'}, L^{'})$ is an ample pair over $R^{s h}$ (a finite-type $R^{s h}$ -scheme with an ample invertible sheaf) equipped with a descent datum relative to $R \to R^{s h}$ , then there are a finite-type $R$ -scheme $X$ and an ample invertible sheaf $L$ on $X$ with $(X, L)_{R^{s h}} ≅ (X^{'}, L^{'})$ compatibly with the datum.

(b) [group descent] If $H$ is a smooth separated finite-type $R^{s h}$ -group scheme with abelian generic fibre containing a fibre-dense open $U_{R^{s h}}$ whose descent datum is effective, and if the group operations of $H$ are compatible with the canonical descent datum on $U_{R^{s h}}$ , then the compatible group descent datum on $H$ is effective: there is a smooth separated finite-type $R$ -group scheme $G$ with $G_{R^{s h}} ≅ H$ compatibly with the operations and with the descended open $U \subseteq G$ .

(c) [completion] In the commissioned abelian completion situation, where $U_{K}$ is the given abelian variety and $H / R^{s h}$ is the group completion of the strict law on $U_{R^{s h}}$ with a stable fibre-dense open $U$ , the canonical descent datum on $U_{R^{s h}}$ satisfies the triple cocycle and extends uniquely to a group descent datum on $H$ ; this datum is effective by (b), and the descended open is fibre-dense in $G$ .

Facts & Assumptions

Given: AC and DC, a discrete valuation ring $R$ with fraction field $K$ and residue field $k$ , a strict henselization $R^{s h}$ , a smooth separated finite-type $R$ -scheme $U$ with strict $R$ -birational group law, and a group completion $H$ of the strict law on $U_{R^{s h}}$ .

[F1]

The group completion of a strict birational group law over $R^{s h}$ exists as a smooth separated finite-type group scheme containing the law as a fibre-dense open subscheme, is unique up to canonical isomorphism, and the stable fibre-dense open $U$ carries the strict law (Finite translate completion and uniqueness).

[F2]

An ample invertible sheaf has a finite cover by affine nonvanishing section opens. Every positive-power section $s$ has quasi-affine nonvanishing locus: raise the affine-cover sections $t_{i}$ and $s$ to the same degree; on $X_{s}$ the functions $t_{i} / s$ have affine principal nonvanishing loci covering $X_{s}$ . Localization of sections identifies each such locus with the corresponding distinguished open of $Spec Γ (X_{s}, O)$ , so the canonical map is an open immersion. These localization and section-extension statements are Extend a quasi-coherent section after multiplying by a power; ampleness is Absolute ampleness by affine section opens. The completion with abelian generic fibre is quasi-projective and admits an ample invertible sheaf $O (D)$ cut out by a fibre-dense affine-complement divisor, after choosing a fibre-dense affine subopen of the stable open downstairs and pulling it back (Divisor ampleness and quasi-projectivity of group models, Affine codimension-one neighbourhoods and divisors).

[F3]

Effective descent of modules and commutative algebras along faithfully flat maps is available, with the descended object described as the invariants; the same holds for graded algebras and their graded pieces (Faithfully flat descent of modules and affine algebras is effective, Faithfully flat descent of modules and algebras is effective).

[F4]

An fpqc covering morphism is submersive, so images of saturated open subschemes are open and can be tested after base change; compatible morphisms between the quasi-compact quasi-separated schemes used here descend along fpqc covers by the affine-cover argument in step 3.1; quasi-compact quasi-affine schemes embed as open subschemes of the spectra of their global-section algebras, retaining the open subscheme as part of the construction, and finite type, separatedness and flatness descend; smoothness will be checked from finite presentation and geometric regularity, and open immersions will be descended as stable open subschemes (Fpqc covers are universally submersive, Fpqc descent of properness components, Flatness descends along faithfully flat base change, Field tests for geometric regularity).

[F5]

Morphisms from a reduced source to a separated target agree if they agree on a schematically dense open. Rational maps descend along the faithfully flat smooth source maps in the cited interface; this is distinct from the fpqc base-extension morphism descent proved in step 3.1 (An S-rational map defined after a faithfully flat smooth base change is defined, Agreement on a schematically dense open, S-dense open subschemes and S-rational maps).

Proof

technique · direct: descend the graded section algebra of an ample pair, glue the descended quasi-affine opens, then descend the group operations as compatible morphisms and use uniqueness of completions for the cocycle

1.1F2F3givenconstruct

For an ample pair $(X^{'}, L^{'})$ with its compatible pair descent datum, form the graded section algebra $B = ⨁_{n \geq 0} Γ (X^{'}, L^{' \otimes n})$ . Flat base change of sections on a finite affine cover and its intersections follows by tensoring their equalizer; hence the pair datum induces compatible descent data on each graded piece and on multiplication. Ampleness gives a finite cover of $X^{'}$ by nonvanishing opens of positive-degree sections, each quasi-affine. These opens, rather than an asserted identity with $Spec B$ , will construct the descended scheme.

2.1F3step 1.1algebra

By effective affine algebra descent [F3] applied to the graded pieces, the datum descends $B$ to a graded $R$ -algebra $B_{0}$ with $R^{s h} \otimes_{R} B_{0} ≅ B$ compatibly with the datum. Every section of $L^{' \otimes n}$ over $R^{s h}$ is a finite sum of $R^{s h}$ -multiples of descended sections of the same degree, because the tensor identification is an isomorphism of graded modules; hence if a section generates $L^{'}$ at a point, at least one descended section does too.

3.1F3F4step 2.1construct

Here is the required morphism descent for the faithfully flat quasi-compact base extension $R \to R^{s h}$ , without a local finite presentation assumption on that extension. Given a compatible morphism $v^{'} : P_{R^{s h}} \to Q_{R^{s h}}$ between descended schemes, cover $Q$ by affine opens $W$ . The opens $(v^{'})^{- 1} (W_{R^{s h}})$ are saturated under the cover's kernel pair because the two pullbacks of $v^{'}$ agree. Their images are open in $P$ by fpqc submersivity, and saturation makes their pullbacks exactly the original opens. On an affine open $V$ in such an image, $v^{'}$ corresponds to a ring map $Γ (W, O) \to Γ (V, O) \otimes_{R} R^{s h}$ . Compatibility puts its image in the faithfully flat equalizer $Γ (V, O)$ , by [F3]; it therefore descends a unique morphism $V \to W$ . The descended maps agree on overlaps because their pullbacks agree, and glue to $v : P \to Q$ . This also descends isomorphisms by descending their inverses. A compatible open immersion is a stable open upstairs, whose open image descends by the same saturated-open argument, and the induced isomorphism descends as just proved. No fppf assertion is applied to $R \to R^{s h}$ .

4.1F3F4step 2.1construct

For each descended homogeneous section $s$ of positive degree, its nonvanishing locus $D (s) \subseteq X^{'}$ is a quasi-affine open subscheme stable under the descent datum, and it descends: embed $D (s)$ into $Spec$ of its global-section algebra, descend that algebra by [F3], and take the image of this stable open in the descended spectrum under the faithfully flat spectrum map; stability makes the inverse image of the image exactly $D (s)$ , and submersivity [F4] makes the image open. These descended opens glue compatibly because compatible morphisms and open immersions descend by step 3.1, producing a finite-type $R$ -scheme $X$ ; the invertible sheaf descends on each affine chart by module descent and glues by uniqueness of descent, giving $L$ on $X$ with $(X, L)_{R^{s h}} ≅ (X^{'}, L^{'})$ . To verify ampleness downstairs, cover each quasi-affine $X_{s}$ by distinguished affine opens of its global-section spectrum contained in $X_{s}$ . Their defining functions extend after multiplication by powers of $s$ by [F2]; multiplying once more by $s$ makes their global nonvanishing loci lie inside $X_{s}$ , where they are exactly those affine opens. These affine section loci cover $X$ , so $L$ is ample by its definition. This proves (a).

5.1F2F4F5step 4.1construct

In (b), choose a fibre-dense affine open $V \subseteq U$ downstairs using the codimension-one neighbourhood lemma in [F2]. Its pullback $V^{'} \subseteq H$ is stable under the descent datum. On the regular smooth $R^{s h}$ -scheme $H$ , its reduced complement is an effective Cartier divisor $D$ with no vertical components by [F2]; equivalently it is the closure of its generic boundary. Stability of $V^{'}$ makes $D$ and $O (D)$ stable with their canonical pair cocycle. Since $H$ has abelian generic fibre and $V^{'}$ is affine and fibre-dense, the ampleness theorem in [F2] makes $O (D)$ ample. Apply (a) to this ample pair to obtain the scheme $G$ with its given descent datum. Multiplication, inverse and unit now descend as compatible morphisms by step 3.1, and their group identities hold downstairs since they hold after the faithfully flat extension. The original stable open $U_{R^{s h}}$ descends to the specified $U \subseteq G$ by morphism and open-immersion descent.

6.1F4step 5.1algebra

The descended $G$ is finite type and separated by fpqc descent of these properties [F4], and flat over $R$ because $H$ is flat and flatness descends [F4]; since $R$ is Noetherian and $G$ is finite type, $G$ is locally of finite presentation. For each residue field, geometric regularity of the fibres descends along the faithfully flat map $R \to R^{s h}$ by [F4]; the smoothness criterion now gives that $G$ is smooth over $R$ . The descended open $U \subseteq G$ is open by descent of open immersions and is fibre-dense because fibre-density is checked after the faithfully flat base change and stabilized by the datum.

7.1F1F5step 5.1step 6.1algebra∎

Finally consider the case (c): $H$ is the group completion of the strict law on $U_{R^{s h}}$ , and $U$ is the stable fibre-dense open model of the given abelian generic variety. Its generic completion is that abelian variety, so $H$ has the abelian generic fibre required in (b). Over each of the two pullbacks $R^{s h} \otimes_{R} R^{s h}$ and the triple tensor product, the pulled-back completions solve the same strict birational law; by the uniqueness part of [F1] they are canonically isomorphic, so the canonical isomorphism on $U$ extends uniquely to a group isomorphism of completions and the triple cocycle holds automatically since the triple-pullback completion is unique. This extends the datum on $U_{R^{s h}}$ uniquely to a group descent datum on $H$ , which is effective by (b); the descended open remains fibre-dense by step 6.1.

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Polarizations and ampleness under Picard twists

Statement

Assume AC and DC as inherited from the supplied scheme and cohomology results. Let $A$ be an abelian variety over a field $k$ and let $L$ be an ample invertible sheaf on $A$ (Absolute ampleness by affine section opens). Then:

(a) the Mumford map $φ_{L} : A \to A^{\lor}$ is a symmetric isogeny, and the bundle $(id, φ_{L})^{*} P$ is ample;

(b) every abelian variety admits a polarization (Polarizations and the Mumford isogeny attached to an ample line bundle);

(c) if $M$ is algebraically trivial (a class in ${Pic}^{0}$ ) then $L \otimes M$ is ample, so ampleness is invariant under twists by algebraically trivial bundles;

(d) for a symmetric homomorphism $λ = φ_{L}$ the bundle $(id, λ)^{*} P$ is ample if and only if $L$ is ample.

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ over a field $k$ , an ample invertible sheaf $L$ on $A$ , and the normalized Poincare bundle $P$ on $A \times_{k} A^{\lor}$ .

[F1]

The Mumford map is a homomorphism, $\ker φ = {Pic}^{0}$ as a sheaf on all tests, and over an algebraic closure every algebraically trivial bundle is of the form $t_{x}^{*} H \otimes H^{- 1}$ for a fixed ample $H$ (Homogeneous bundles and Mumford surjectivity). Over an algebraic closure an ample bundle gives a Mumford isogeny with finite scheme-theoretic kernel (Coherent Kunneth, the tangent bound and the proper-image dual, Statement (c)); the dual identifies with the base change of $A^{\lor}$ (Finite-field descent of the dual and the Poincare bundle). A proper quasi-finite morphism is finite (A proper quasi-finite morphism is finite).

[F2]

Under biduality, $κ_{A}$ classifies the switched normalized Poincare bundle, and dualizing a homomorphism pulls back line bundles (Dual isogenies, Cartier-dual kernels and canonical biduality). The normalized square $Λ (L)$ is invariant under exchanging its two $A$ -factors (Polarizations and the Mumford isogeny attached to an ample line bundle). These descriptions permit the symmetry comparison in step 1.1; the lemma on symmetric homomorphisms does not supply symmetry as a premise.

[F3]

$(id \times φ_{L})^{*} P ≅ Λ (L) = m^{*} L \otimes p_{1}^{*} L^{- 1} \otimes p_{2}^{*} L^{- 1} \otimes π^{*} e^{*} L$ , whence $φ_{(id, φ_{L})^{*} P} = 2 φ_{L} = φ_{L^{2}}$ , so the two bundles differ by an algebraically trivial class (Polarizations and the Mumford isogeny attached to an ample line bundle, Homogeneous bundles and Mumford surjectivity).

[F4]

Every abelian variety is projective, positive powers of ample bundles are ample and sufficiently high powers are very ample, ample pulls back along finite morphisms, the external Segre tensor of ample bundles is ample, and ampleness descends under field extension (Ampleness of a given line bundle descends under field extension; Every abelian variety over a field is projective, High powers of an ample line bundle embed a proper scheme, Ampleness is invariant under positive powers, Finite pullback preserves absolute ampleness, Segre embedding and its line bundle, Global functions on proper integral schemes form a finite extension of the base field).

Proof

technique · direct: compare the bundle $(\operatorname{id},\varphi_{\mathcal L})^*\mathcal P$ with $\mathcal L^2$, use that both have Mumford map $2\varphi_{\mathcal L}$, and invoke invariance of ampleness under algebraically trivial twists

1.1F1F2F3givenconstruct

By [F1], $φ_{L, \overset{ˉ}{k}}$ is an isogeny with finite scheme-theoretic kernel. Every geometric fibre of $φ_{L}$ is, after choosing a point in it, a translate of that geometric kernel. In particular $K (L) \to Spec k$ is proper and quasi-finite, since it is closed in $A$ and has a finite geometric fibre; [F1] makes it finite over $k$ . Surjectivity follows from geometric surjectivity after the field extension, so $φ_{L}$ is an isogeny. By [F3], $(id, φ_{L})^{*} P$ has Mumford map $2 φ_{L}$ , the same as $L^{2}$ ; since $\ker φ = {Pic}^{0}$ as a sheaf [F1], the two bundles differ by an algebraically trivial class: $(id, φ_{L})^{*} P ≅ L^{2} \otimes M$ with $M \in {Pic}^{0}$ . For symmetry, the family classifying $φ_{L}^{\lor} \circ κ_{A}$ on the second copy of $A$ is obtained by switching the Poincare factors and pulling back along $φ_{L}$ on the other factor. It is therefore the switched bundle $σ^{*} Λ (L)$ , where $σ (x, y) = (y, x)$ . The family classifying $φ_{L}$ is $Λ (L)$ . Since multiplication is commutative, the two normalized bundles are isomorphic, including their rigidifications on both axes. The all-test universal property gives $φ_{L}^{\lor} \circ κ_{A} = φ_{L}$ , which proves symmetry rather than assuming it. For the bundle comparison in [F3], diagonal pullback of $Λ (L)$ gives $[2]^{*} L \otimes L^{- 2}$ up to a constant line. For any homomorphism $f$ , translation commutation and pullback of Picard classes give $φ_{f^{*} L} = f^{\lor} φ_{L} f$ ; with $f = [2]$ and additivity of duality this gives $φ_{[2]^{*} L} = 4 φ_{L}$ and hence the asserted $2 φ_{L}$ .

1.2F1F4givenalgebra

We first prove (c). Let $M \in {Pic}^{0} (A)$ and let $H$ be ample. Over an algebraic closure, [F1] gives $M ≅ t_{x}^{*} H \otimes H^{- 1}$ for some point $x$ , so $H \otimes M ≅ t_{x}^{*} H$ is the pullback of an ample bundle under an isomorphism, hence ample. Ampleness is a geometric condition checked after faithfully flat field extension, so $L \otimes M$ is ample over $k$ ; this is (c).

2.1F4step 1.1step 1.2algebra

Statement (a) now follows: since $L^{2}$ is ample and $(id, φ_{L})^{*} P$ differs from it by the algebraically trivial class $M$ of step 1.1, (c) gives that $(id, φ_{L})^{*} P$ is ample.

3.1F3F4step 1.1step 2.1construct

For (b), $A$ is projective by [F4], so there exists an ample invertible sheaf $L$ on $A$ ; then $φ_{L}$ is a symmetric isogeny by (a) and $(id, φ_{L})^{*} P$ is ample. Since $φ_{L}$ is the Mumford map of an ample bundle already over $k$ , it satisfies the geometric ample-realization definition of a polarization.

4.1F1F3F4step 1.2algebra∎

For (d), let $λ = φ_{L}$ be symmetric, now allowing $L$ to be any invertible sheaf, and put $N = (id, λ)^{*} P$ . The identity [F3], which holds without ampleness, gives $φ_{N} = 2 φ_{L} = φ_{L^{2}}$ ; by [F1] this means $N ≅ L^{2} \otimes M$ for an algebraically trivial $M$ . If $L$ is ample, then $L^{2}$ is ample by [F4] and $N$ is ample by step 1.2. Conversely, if $N$ is ample, applying step 1.2 to $N$ and $M^{- 1}$ makes $L^{2}$ ample, and [F4] then makes $L$ ample.

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The dual abelian variety, the Poincare bundle and polarizations

Statement

Assume AC and DC as inherited from projectivity and the supplied cohomology machinery. Let $A$ be an abelian variety over a field $k$ , of dimension $g$ . Then:

(a) [existence and duality] the degree-zero part of the rigidified relative Picard functor of $A / k$ (The rigidified relative Picard functor and the dual abelian variety) is representable by an abelian variety $\hat{A}$ , the dual abelian variety, of dimension $g$ , with universal Poincare sheaf $P$ on $A \times_{k} \hat{A}$ ; the canonical homomorphism $A \to \hat{\hat{A}}$ is an isomorphism;

(b) [functoriality] $A \mapsto \hat{A}$ is a contravariant functor on abelian varieties over $k$ , and for every isogeny $f : A \to B$ the dual $f^{\lor} : \hat{B} \to \hat{A}$ is an isogeny with kernel the Cartier dual of $\ker f$ and degree $\deg f^{\lor} = \deg f$ ;

(c) [Mumford maps] for every invertible sheaf $L$ on $A$ the Mumford homomorphism $φ_{L} : A \to \hat{A}$ exists; if $L$ is ample then $φ_{L}$ is a symmetric isogeny with finite kernel $K (L)$ ; every symmetric homomorphism $A \to \hat{A}$ is $φ_{L}$ for some invertible sheaf $L$ after base change to a separably closed field;

(d) [polarizations] an ample $L$ makes $φ_{L}$ a polarization, every abelian variety admits a polarization, the degree of a polarization is a perfect square, and $A$ is projective.

Facts & Assumptions

Given: AC and DC, an abelian variety $A$ of dimension $g$ over a field $k$ , and the rigidified relative Picard functor of The rigidified relative Picard functor and the dual abelian variety.

[F1]

The algebraically trivial rigidified Picard subfunctor is represented by an abelian variety $A^{\lor}$ of dimension $g$ with a normalized Poincare bundle on $A \times_{k} A^{\lor}$ , and the formation and universal property are compatible with field extension (Finite-field descent of the dual and the Poincare bundle).

[F2]

Duality is contravariantly functorial on homomorphisms: for composable homomorphisms $f : A \to B$ and $g : B \to C$ , $(g \circ f)^{\lor} = f^{\lor} \circ g^{\lor}$ ; it is additive for parallel homomorphisms $f, g : A \to B$ , so $(f + g)^{\lor} = f^{\lor} + g^{\lor}$ . If $f$ is an isogeny, then $f^{\lor}$ is an isogeny with kernel $(\ker f)^{D}$ and degree $\deg f^{\lor} = \deg f$ . The canonical biduality morphism $κ_{A} : A \to A^{\lor \lor}$ is an isomorphism and is natural in $A$ (Dual isogenies, Cartier-dual kernels and canonical biduality, Normal subgroup quotients of finite-type group schemes exist as fppf scheme quotients).

[F3]

The theorem of the square makes $φ_{L}$ a homomorphism into the degree-zero part for every invertible sheaf $L$ (The theorem of the square and the Mumford homomorphism into the Picard group); ample bundles give symmetric isogenies and every symmetric homomorphism is a Mumford map over a separably closed field, with finite separable realization in general (Polarizations and ampleness under Picard twists, Symmetric homomorphisms are Mumford maps).

[F4]

Every abelian variety is projective and therefore carries an ample invertible sheaf; the degree of any polarization equals $χ (A, L)^{2}$ for an ample $L$ realizing it (Every abelian variety over a field is projective, The square degree of a Mumford map, Polarizations and the Mumford isogeny attached to an ample line bundle).

Proof

technique · direct: assemble the commissioned clauses from the constructed dual, the dual-isogeny calculus, the symmetric-map realization and the square-degree computation

1.1F1F2givenconstruct

Clause (a) is [F1]: the degree-zero rigidified Picard subfunctor is represented by an abelian variety $\hat{A} = A^{\lor}$ of dimension $\dim A = g$ with universal normalized Poincare sheaf $P$ on $A \times_{k} A^{\lor}$ , and the formation is compatible with field extension. The canonical morphism $κ_{A} : A \to A^{\lor \lor}$ is an isomorphism by [F2], which is the biduality statement of (a).

1.2F2givenalgebra

Clause (b) is [F2]: pullback of rigidified bundles defines the dual homomorphism for every homomorphism. Composition is contravariant for composable homomorphisms $f : A \to B$ , $g : B \to C$ , and additivity $(f + g)^{\lor} = f^{\lor} + g^{\lor}$ is for parallel homomorphisms $f, g : A \to B$ . When $f$ is an isogeny, $f^{\lor}$ is an isogeny with kernel $(\ker f)^{D}$ and degree $\deg f$ . Thus $A \mapsto \hat{A}$ is a contravariant functor, and the duality identities used here have the required domains.

1.3F3givenconstruct

Clause (c): for an invertible sheaf $L$ the Mumford map $φ_{L}$ is a homomorphism into $\hat{A}$ by [F3]; if $L$ is ample, [F3] gives that $φ_{L}$ is a symmetric isogeny with finite kernel $K (L)$ . Conversely, if $λ : A \to \hat{A}$ is symmetric, then over a separably closed extension field [F3] realizes $λ$ as $φ_{L}$ for an invertible $L$ ; over an arbitrary field the realization exists after a finite separable extension in general, as stated.

2.1F3F4givenalgebra∎

Clause (d): if $L$ is ample, [F3] shows that $φ_{L}$ is a symmetric isogeny with $(id, φ_{L})^{*} P$ ample, so it is a polarization by definition; every $A$ is projective by [F4] and hence carries an ample $L$ , giving a polarization. For any polarization $λ$ realized by an ample $L$ over an algebraic closure, [F4] gives $\deg λ = χ (A_{\overset{ˉ}{k}}, L)^{2}$ , a perfect square, and the degree is unchanged by field extension. Projectivity of $A$ is [F4].

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Full minimal model embedding

Statement

Assume AC and DC as inherited from the supplied algebra and scheme results. Let $R$ be a discrete valuation ring with fraction field $K$ , residue field $k$ and a chosen strict henselization $R^{s h}$ , and let $A / K$ be an abelian variety. Let $X$ be the smooth separated finite-type faithfully flat $R$ -model of $A$ supplied by Separated minimal union and translations, with strictification $U \subseteq X$ , and let $G$ be the descended group completion over $R$ of the strict law on $U_{R^{s h}}$ (Effective ample-pair and group descent from a strict henselization). Then $G$ contains the full model $X$ , not only its strictification $U$ , as an $R$ -dense open subscheme.

Facts & Assumptions

Given: AC and DC, a discrete valuation ring $R$ with fraction field $K$ and residue field $k$ , a strict henselization $R^{s h}$ , an abelian variety $A / K$ , the separated minimal model $X / R$ of Separated minimal union and translations with strictification $U \subseteq X$ , and the descended group completion $G / R$ containing $U$ as an $R$ -dense open.

[F1]

$X$ is smooth, separated, finite type and faithfully flat over the regular Noetherian base $R$ , integral with generic fibre $A$ , and $U \subseteq X$ is an $R$ -dense (fibre-dense) open subscheme carrying a strict $R$ -birational group law; $U_{R^{s h}}$ is an open subscheme of the smooth separated finite-type $R^{s h}$ -group scheme $H$ , which descends to the smooth separated finite-type $R$ -group scheme $G$ containing $U$ (Separated minimal union and translations, Effective ample-pair and group descent from a strict henselization, S-dense open subschemes and S-rational maps).

[F2]

A rational map from a smooth $S$ -scheme to a smooth separated finite-type $S$ -group scheme over a regular Noetherian base which is defined at every height-one point extends uniquely to an $S$ -morphism (Weil's extension theorem for rational maps into smooth separated group schemes).

[F3]

On a smooth finite-type $R$ -scheme the total space is regular, hence normal and locally factorial, and a nonzero rational section of a line bundle has a Cartier divisor of pure codimension one; a Noetherian normal domain is the intersection of its height-one localizations, so a rational function regular at every height-one point is regular (Regularity ascends and descends along a flat local homomorphism, Locally standard smooth iff flat with geometrically regular fibres, Regular local rings are unique factorization domains, Rational sections of line bundles are Cartier divisors, A normal Noetherian domain is the intersection of its height-one localizations).

[F4]

A smooth $R$ -group scheme has translation-invariant top forms; a morphism between smooth models of equal relative dimension is etale where its top differential is an isomorphism (Invariant volume and finite minimal classes, Dilatations and defect computation). Two morphisms from a reduced source to a separated target agree on the entire source if they agree on a schematically dense open (Agreement on a schematically dense open). Descent additionally requires a faithfully flat cover and equality of the two pullbacks; no descent is inferred from reducedness or separatedness alone.

[F5]

Zariski's main factorization: a separated quasi-finite morphism to a quasi-compact base factors as an open immersion followed by a finite morphism, locally on the base (Scheme Zariski Main factorization for separated quasi-finite morphisms).

Proof

technique · direct: extend the identity rational map $X\dashrightarrow G$ using the codimension-one criterion, show its invariant volume is a unit so that it is etale, and conclude by the scheme Hartogs and Zariski-factorization argument that it is an open immersion

1.1F1givenconstruct

The generic fibre of $X$ is $A = X_{K}$ , and $G_{K}$ is the group completion of the strict law on $U_{K} = A$ ; hence the identity of $A$ defines a rational map $φ : X ⇢ G$ over $R$ which on $U$ is the given open immersion $U ↪ G$ . Every height-one point of $X$ lies either in the generic fibre (where $φ$ is defined, since it is the identity of $A$ ) or is a generic point of an irreducible component of the special fibre. Since $U$ is $R$ -dense, its complement contains no irreducible component of any fibre, so $U$ contains the generic point of every component of the special fibre; therefore $φ$ is defined at every height-one point of $X$ .

2.1F1F2step 1.1construct

The base $R$ is a regular Noetherian scheme, $X$ is smooth over $R$ and $G$ is a smooth separated finite-type $R$ -group scheme, so the codimension-one extension criterion [F2] applies to $φ$ and produces a unique $R$ -morphism $ψ : X \to G$ extending $φ$ . On the generic fibre $ψ_{K}$ is the identity of $A$ , so $ψ$ is birational.

3.1F3F4step 1.1step 2.1algebra

We show that $ψ$ is etale. Its top differential is a section of the invertible sheaf $Hom (ψ^{*} ⋀^{g} Ω_{G / R}, ⋀^{g} Ω_{X / R})$ . It is a unit on $U$ , where $ψ$ is the given open immersion, and on the generic fibre, where it is the identity. Thus it is nonzero generically, and its zero divisor can only be supported in $X ∖ U$ . Step 1.1 shows that $U$ contains every height-one point. Since $X$ is regular and integral, the zero locus of a nonzero section of an invertible sheaf is an effective Cartier divisor of pure codimension one, unless empty. There is no possible codimension-one support, so the section is nowhere zero and the top differential is an isomorphism. The differential criterion [F4] makes $ψ$ etale. This argument requires no global trivialization of the canonical bundle of the preliminary model.

4.1F3F5step 3.1algebra

The morphism $ψ$ is separated, because $X$ and $G$ are separated over $R$ , and it is quasi-finite: it is etale, hence locally quasi-finite, and it is a morphism of finite type between quasi-compact schemes, hence quasi-compact; a quasi-compact locally quasi-finite morphism is quasi-finite. Being etale and birational, and having reduced integral source and normal target (smooth over the DVR), it is an open immersion: apply Zariski's main factorization [F5] locally on $G$ to write $ψ = g \circ j$ with $j : X ↪ Z$ an open immersion and $g : Z \to G$ finite. Replace $Z$ by the reduced closure of its birational generic component, which still contains $j (X)$ . Its coordinate algebra is a finite integral subalgebra of the common function field containing the normal target algebra, so it equals that target algebra. Thus $g$ is an isomorphism on this component and $ψ$ is an open immersion.

5.1F1step 4.1givenalgebra∎

Consequently $ψ$ identifies $X$ with an open subscheme of $G$ containing $U$ ; since the completion already contains $U$ as an $R$ -dense open by [F1], the larger image of $X$ is $R$ -dense in $G$ . This is the assertion that the descended completion contains the full separated minimal model $X$ , not only the strictification $U$ , as an $R$ -dense open.

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Existence of Neron models for abelian varieties over a discrete valuation ring

Statement

Assume AC and DC. Let $R$ be an arbitrary discrete valuation ring with fraction field $K$ and residue field $k$ , and let $A_{K}$ be an abelian variety over $K$ . Then there exists a smooth separated finite-type $R$ -group scheme $N$ with generic fibre $A_{K}$ such that for every smooth $R$ -scheme $Z$ restriction ${Hom}_{R} (Z, N) ⟶ {Hom}_{K} (Z_{K}, A_{K})$ is bijective. The model $N$ is unique up to a unique isomorphism inducing the given identity on generic fibres. No excellence, completeness, perfect-residue-field or reduction-type hypothesis is imposed.

Facts & Assumptions

Given: AC and DC, an arbitrary discrete valuation ring $R$ with fraction field $K$ and residue field $k$ , and an abelian variety $A_{K}$ over $K$ .

[F1]

There exists a smooth separated finite-type faithfully flat $R$ -model $X$ of $A_{K}$ carrying a birational group law with birational universal translations; the law restricts to a strict law on an $R$ -dense model open $U \subseteq X$ , whose multiplication domain is an open of $U \times_{R} U$ with strict universal translations, given by graph closures (Separated minimal union and translations, Birational group law, Strictification, Strict law graph calculus).

[F2]

Over a strict henselization $R^{s h}$ , finitely many section translates complete the strict law to a smooth separated finite-type $R^{s h}$ -group scheme $H$ containing $U_{R^{s h}}$ as a fibre-dense open, uniquely; the canonical descent datum on this completion is effective and gives a smooth separated finite-type $R$ -group scheme $G$ containing $U$ , and the full model $X$ embeds in $G$ as an $R$ -dense open (Finite translate completion and uniqueness, Effective ample-pair and group descent from a strict henselization, Full minimal model embedding).

[F3]

If $S$ is a regular Noetherian base, $Z$ smooth over $S$ , $G$ a smooth separated finite-type $S$ -group scheme, and an $S$ -rational map $Z ⇢ G$ is defined at every height-one point of $Z$ , then it extends uniquely to an $S$ -morphism $Z \to G$ (Weil's extension theorem for rational maps into smooth separated group schemes).

[F4]

Domains of $R$ -rational maps are fibre-dense; morphisms into a separated target that agree on a schematically dense open agree everywhere (S-dense open subschemes and S-rational maps, Agreement on a schematically dense open).

[F5]

Morphisms descend along faithfully flat, quasi-compact, locally finitely presented covers when the two pullbacks agree (Scheme morphisms satisfy fppf descent, Faithfully flat scheme morphism). A finitely presented open neighbourhood and morphism over a filtered-colimit local ring spread to a finite stage (Finite-stage descent of finitely presented schemes and their morphisms).

[F6]

Two smooth separated finite-type $R$ -models of $A_{K}$ satisfying the extension property are uniquely isomorphic over $R$ compatibly with their specified generic-fibre identifications; Neron models over Dedekind bases are compatible with etale base change (Uniqueness, weak Neron property, etale base change and local nature of Neron models, Neron models, the Neron mapping property and weak Neron models).

Proof

technique · build the group model from the minimal model, strictification and effective completion. For the mapping property, extend the generic translation on the minimal model by the codimension-one Weil criterion and descend its value using the faithfully flat model cover

1.1F1F2givenconstruct

By [F1] construct the separated minimal model $X$ of $A_{K}$ , its strictification $U$ , and finally by [F2] the descended smooth separated finite-type $R$ -group scheme $G$ with generic fibre $G_{K} = A_{K}$ and $X \subseteq G$ as an $R$ -dense open.

1.2F1F2F3F4F5givenconstruct

To prove the mapping property, let $Z$ be a smooth $R$ -scheme and $u_{K} : Z_{K} \to A_{K}$ a $K$ -morphism. Work first with $Z$ of finite type; arbitrary $Z$ is covered by finite-type opens and the unique extensions glue. Put $Y = Z \times_{R} X$ . On its generic fibre define $θ_{K} : Y_{K} \to G_{K}$ by $θ_{K} (z, x) = u_{K} (z) x$ , using the group law of $G_{K} = A_{K}$ . For each generic point $η$ of an irreducible component of $Z_{k}$ , the local ring $R^{'} = O_{Z, η}$ is a DVR, with fraction field $K^{'}$ , and restriction of $u_{K}$ along $Spec K^{'} \to Z_{K}$ gives a point of $A_{K} (K^{'})$ . The translation supplier Separated minimal union and translations extends translation by this point to an $R^{'}$ -birational self-map of $X_{R^{'}}$ which is an open immersion on its $R^{'}$ -dense domain. This domain contains the generic point of every component of the special fibre of $X_{R^{'}}$ , so $θ_{K}$ extends at the corresponding generic points of $Y_{k}$ . Since $R^{'}$ is the filtered colimit of the rings of affine neighbourhoods of $η$ , finite-presentation descent [F5] spreads a quasi-compact open neighbourhood and its morphism to $G$ to a neighbourhood in $Y$ of each such point. There are finitely many vertical generic points. Together with the generic fibre, these neighbourhoods form an $R$ -dense open in $Y$ ; the local maps agree on overlaps because the generic fibre is schematically dense in each overlap and $G$ is separated ([F4]), so they glue to an $R$ -rational map $θ : Y ⇢ G$ . It is defined at every height-one point of $Y$ : the horizontal ones lie in $Y_{K}$ , and each vertical height-one point is the generic point of a component of $Y_{k}$ just treated. Since $Y$ is smooth over the regular Noetherian DVR and $G$ is a smooth separated finite-type group scheme, [F3] extends $θ$ uniquely to a morphism $Θ : Y \to G$ .

2.1F2F4F5step 1.2algebra

Let $j : X ↪ G$ be the dense open embedding from [F2], and let $ι_{G}$ and $m_{G}$ be inversion and multiplication on $G$ . Define $h : Y = Z \times_{R} X \to G$ by $h (z, x) = m_{G} (Θ (z, x), ι_{G} (j (x))) .$ On $Y_{K}$ this is $u_{K} (z) x x^{- 1} = u_{K} (z)$ , independent of $x$ . The projection $p : Y \to Z$ is faithfully flat, quasi-compact and locally finitely presented because $X \to Spec R$ is smooth, finite type and faithfully flat. On $Y \times_{Z} Y = Z \times_{R} X \times_{R} X$ , the two pullbacks of $h$ agree on the generic fibre, hence everywhere by [F4] and separatedness of $G$ . Fppf descent [F5] therefore gives a unique $R$ -morphism $u : Z \to G$ with $u \circ p = h$ ; its generic fibre is $u_{K}$ . Uniqueness follows because any two extensions agree on the schematically dense generic fibre and $G$ is separated. This proves existence and uniqueness of the extension for all smooth $Z$ .

3.1F4F6step 2.1algebra

Uniqueness of the model: if $N$ and $N^{'}$ both satisfy the extension property with generic fibre $A_{K}$ , then the identity of $A_{K}$ extends to $R$ -morphisms $N \to N^{'}$ and $N^{'} \to N$ . Both composites extend the identity of $A_{K}$ , hence equal the respective identities by the uniqueness clause of the extension property (applied to the models themselves); so $N ≅ N^{'}$ uniquely, and [F4] identifies the canonical isomorphism.

4.1F1F2F4step 1.1step 2.1algebra∎

The construction used only an arbitrary discrete valuation ring: the minimal model, strictification, strict-henselian completion, effective group descent and Weil extension all hold without excellence, completeness, perfect residue field or any restriction on the reduction type, so the stated class is exactly as claimed.

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The Neron-Ogg-Shafarevich criterion in residue characteristic prime to l

Statement

Assume AC and DC as inherited from the stated suppliers. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$ , let $A / K$ be an abelian variety of dimension $g$ with finite-type Neron model $N / R$ (Existence of Neron models for abelian varieties over a discrete valuation ring, Neron models, the Neron mapping property and weak Neron models), and let $ℓ \neq char k$ be a prime. Then the following are equivalent:

(a) $A$ has good reduction over $R$ , i.e. there is an abelian scheme over $R$ with generic fibre $A$ (Good reduction of an abelian variety over a Dedekind scheme);

(b) the Neron model $N$ is an abelian scheme over $R$ ;

(c) all the torsion groups $A [ℓ^{ν}] (K^{s e p})$ , $ν \geq 1$ , are fixed pointwise by the inertia group $I \subseteq Gal (K^{s e p} / K)$ ;

(d) the Tate module $T_{ℓ} A$ is unramified at $R$ (Prime-to-residue-characteristic Tate modules and inertia).

Facts & Assumptions

Given: AC and DC, a discrete valuation ring $R$ with fraction field $K$ , residue field $k$ and strict henselization $R^{s h}$ , an abelian variety $A / K$ of dimension $g$ , its finite-type Neron model $N / R$ , and a prime $ℓ \neq char k$ .

[F1]

An abelian scheme over $R$ with generic fibre $A$ is a Neron model of $A$ , and Neron models of smooth separated finite-type $K$ -schemes are unique up to a unique $R$ -isomorphism inducing the identity on generic fibres (An abelian scheme is the Neron model of its generic fibre, Uniqueness, weak Neron property, etale base change and local nature of Neron models).

[F2]

For an abelian scheme $B / R$ of relative dimension $g$ and $ℓ \neq char k$ , each $B [ℓ^{ν}]$ is finite etale over $R$ of rank $ℓ^{2 g ν}$ , and over $R^{s h}$ specialization identifies its geometric generic points with its special separable points; the inertia group acts trivially on $B [ℓ^{ν}]$ and on the Tate module (Abelian scheme torsion specialization is unramified, Prime to characteristic multiplication is etale).

[F3]

For a field $F$ and $ℓ \neq char F$ , $A [ℓ^{ν}] (F^{s e p}) ≅ (Z / ℓ^{ν})^{2 g}$ and $T_{ℓ} (A)$ is free of rank $2 g$ ; an automorphism of $F^{s e p}$ over $F$ acts trivially on $T_{ℓ} (A)$ if and only if it acts trivially on every $A [ℓ^{ν}] (F^{s e p})$ (Field prime to characteristic torsion and Tate module, Prime-to-residue-characteristic Tate modules and inertia).

[F4]

Over a strictly henselian local ring with separably closed residue field, a separated etale finite-type scheme $H$ satisfies $H (R) ≅ H (k)$ by reduction. A Neron model has the extension property for etale local points; for $R^{s h}$ this follows by finite-stage approximation, and separatedness makes $N (R^{s h}) \to A (K^{s h})$ bijective (Strict henselian etale sections, Uniqueness, weak Neron property, etale base change and local nature of Neron models, Neron models, the Neron mapping property and weak Neron models).

[F5]

If $G$ is a smooth commutative finite-type group scheme of dimension $g$ over a field of characteristic $\neq ℓ$ with $∣ G [ℓ^{ν}] (\overset{ˉ}{k}) ∣ = ℓ^{2 g ν}$ for every $ν \geq 1$ , then $G^{0}$ is an abelian variety; a smooth separated finite-type quasi-projective $R$ -scheme with abelian generic fibre and proper geometrically connected special fibre is proper over $R$ and an abelian scheme on its identity component; a smooth group scheme over a discrete valuation ring has an open identity component with connected generic and special fibres (Special fibre torsion growth detects properness, Connected smooth quasiprojective model with proper special fibre is proper, Divisor ampleness and quasi-projectivity of group models, The identity model of a smooth group with abelian generic fibre).

[F6]

The Neron group scheme $N$ is commutative: its generic group law is commutative, and the two multiplication morphisms $N \times_{R} N \to N$ agree on the schematically dense generic fibre because $N$ is separated (Agreement on a schematically dense open). For a smooth commutative $R$ -group scheme, if $n$ is a unit then $[n] : N \to N$ is etale: its differential at the identity is multiplication by $n$ on the locally free Lie module, and translations identify the differential at every point; apply the equal-relative-dimension criterion (Group schemes over a base scheme, Differentials of a smooth morphism, Dilatations and defect computation). Its kernel is therefore a separated etale finite-type $R$ -scheme, though it need not be finite.

Proof

technique · direct: reduce good reduction to the Neron model, identify special-fibre torsion through etale sections over the strict henselization, and detect properness of the identity component by prime-to-characteristic torsion growth

1.1F1givenalgebra

(a) $\Leftrightarrow$ (b). If $A$ has good reduction with abelian scheme model $B$ , then $B$ is a Neron model of $A$ by [F1], so $B ≅ N$ by uniqueness and $N$ is an abelian scheme. Conversely if $N$ is an abelian scheme over $R$ with generic fibre $N_{K} ≅ A$ , then $N$ is an abelian scheme model of $A$ , i.e. (a) holds.

1.2F3givenalgebra

(c) $\Leftrightarrow$ (d). By definition of the inertia group and of the unramified Tate module [F3], $I$ acts trivially on $T_{ℓ} A = {\underset{\leftarrow}{l i m}}_{ν} A [ℓ^{ν}] (K^{s e p})$ if and only if it acts trivially on every finite quotient $A [ℓ^{ν}] (K^{s e p})$ , which is precisely (c).

1.3F3F4F6construct

(c) $\Rightarrow$ (b). Assume all $ℓ^{ν}$ -torsion is inertia-fixed. By [F6], multiplication by $ℓ^{ν}$ on the smooth group scheme $N$ is etale, so its kernel $N [ℓ^{ν}]$ is a separated etale finite-type $R$ -scheme. We do not need this kernel to be finite: over $R^{s h}$ , [F4] gives the reduction bijection $N [ℓ^{ν}] (R^{s h}) \tilde{\to} N_{k} [ℓ^{ν}] (k^{s e p}) .$ The weak Neron property and separatedness identify $N (R^{s h})$ with $A (K^{s h})$ , compatibly with the group laws. Therefore $N_{k} [ℓ^{ν}] (k^{s e p}) ≅ A (K^{s h}) [ℓ^{ν}] = A [ℓ^{ν}] (K^{s e p})^{I} = A [ℓ^{ν}] (K^{s e p}),$ where $K^{s h} = (K^{s e p})^{I}$ and the last equality is hypothesis (c). By [F3], this set has cardinality $ℓ^{2 g ν}$ . Now apply [F5] to the smooth commutative finite-type special fibre $N_{k}$ of dimension $g$ : its identity component $N_{k}^{0}$ is an abelian variety, hence proper.

2.1F2step 1.1algebra

(b) $\Rightarrow$ (c). If $N$ is an abelian scheme, [F2] gives that each $N [ℓ^{ν}]$ is finite etale of rank $ℓ^{2 g ν}$ over $R$ and that inertia acts trivially on the geometric torsion points; under the identification $A [ℓ^{ν}] (K^{s e p}) = N [ℓ^{ν}] (K^{s e p})$ this is exactly the pointwise invariance of (c).

2.2F1F5step 1.3algebra

The identity component $N^{0}$ is an open smooth separated finite-type $R$ -subgroup scheme with generic fibre $A$ and geometrically connected special fibre $N_{k}^{0}$ [F5]. It is quasi-projective by [F5], so the connected-model properness criterion makes it proper over $R$ ; a smooth proper $R$ -group scheme with abelian generic fibre of dimension $g$ is an abelian scheme. By [F1], this abelian scheme $N^{0}$ is a Neron model of $A$ . Both $N$ and $N^{0}$ are now Neron models of the same generic fibre, so uniqueness [F1] identifies them; hence $N$ is an abelian scheme, proving (b).

3.1step 1.1step 2.1step 1.2step 2.2algebra∎

The implications (a) $\Leftrightarrow$ (b) (step 1.1), (b) $\Rightarrow$ (c) (step 2.1), (c) $\Leftrightarrow$ (d) (step 1.2) and (c) $\Rightarrow$ (b) (steps 1.3 and 2.2) close the cycle, proving the equivalence of (a)–(d).

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Good reduction, coherent base change, and unramified torsion

Statement

Assume the Axiom of Choice and the Axiom of Dependent Choice, inherited from the cohomology-and-base-change suppliers. Let $S$ be a Dedekind scheme with function field $K$ , let $f : A \to S$ be an abelian scheme of relative dimension $g$ , and let $A_{K}$ be its generic fibre. Then:

(a) [good reduction] $A_{K}$ is an abelian variety with good reduction over $S$ , the abelian scheme model $A$ is unique up to a unique $S$ -isomorphism inducing the specified identity on $A_{K}$ , and $A$ is the Neron model of $A_{K}$ ; for every closed point $s \in S$ the fibre $A_{s}$ is an abelian variety of dimension $g$ over $κ (s)$ .

(b) [base change] For every morphism $S^{'} \to S$ the base change $A_{S^{'}} \to S^{'}$ is an abelian scheme of relative dimension $g$ , it represents the base change of the good reduction data, and if $S^{'}$ is a Dedekind scheme with function field $K^{'}$ and $S^{'} \to S$ is dominant so that $K \to K^{'}$ is defined, then $A_{K^{'}}$ has good reduction over $S^{'}$ .

(c) [coherent cohomology and base change] Fix $s \in S$ , $q \geq 0$ and a coherent $O_{A}$ -module $F$ flat over $S$ , and let $φ_{s}^{q} : (R^{q} f_{*} F) (s) \to H^{q} (A_{s}, F_{s})$ be the cohomology-and-base-change map. If $φ_{s}^{q}$ is surjective, then there is an affine open neighbourhood $U \subseteq S$ of $s$ such that for every $T \to U$ the base-change map $h^{*} (R^{q} f_{*} F ∣_{U}) \to R^{q} f_{T *}^{'} (g_{T}^{*} F)$ is an isomorphism; if moreover $φ_{s}^{q - 1}$ is surjective, then $R^{q} f_{*} F$ is finite locally free on a neighbourhood of $s$ . For $q = 0$ and $F = O_{A}$ the unit map $O_{S} \to f_{*} O_{A}$ is an isomorphism, with inverse evaluation along the identity section, and for $F$ a line bundle flat over $S$ the higher direct images are finite locally free wherever the successive base-change maps are surjective.

(d) [residue restrictions] No restriction on the residue characteristic of $S$ is imposed in (a)-(c); the finite flatness conclusions in (c) are stated under the exact surjectivity hypotheses of the cohomology-and-base-change theorem, because fibrewise cohomology of smooth proper families need not be locally constant in residue characteristic $p > 0$ .

(e) [prime-to-residue-characteristic etale good reduction] Locally let $R$ be a discrete valuation ring, $K$ its fraction field and $k$ its residue field, $A_{K}$ an abelian variety and $ℓ$ a prime different from $char k$ . Then $A_{K}$ has good reduction over $R$ if and only if inertia acts trivially on $T_{ℓ} (A_{K})$ , equivalently on $A_{K} [ℓ^{ν}] (K^{s e p})$ for every $ν \geq 1$ .

Facts & Assumptions

Given: AC and DC, a Dedekind scheme $S$ with function field $K$ , an abelian scheme $f : A \to S$ of relative dimension $g$ with generic fibre $A_{K}$ , and the base-change maps of Cohomology and base-change map.

[F1]

The generic fibre of an abelian scheme is an abelian variety of the same dimension, fibres of abelian schemes are abelian varieties, and base change of an abelian scheme is an abelian scheme of the same relative dimension (Abelian schemes over a base, Abelian varieties over a field, Base change and products of abelian schemes).

[F2]

An abelian scheme over a Dedekind scheme is the Neron model of its generic fibre, is unique as an abelian scheme model up to a unique isomorphism inducing the specified identity on the generic fibre, and every abelian variety with a good-reduction model admits a Neron model (An abelian scheme is the Neron model of its generic fibre, Good reduction supplies a Neron model, Good reduction is stable under base change of the base).

[F3]

Cohomology and base change for a proper finite-presentation morphism and a coherent sheaf flat over the base gives the following conclusions: surjectivity of the base-change map at a point propagates to an isomorphism on an affine neighbourhood for arbitrary test bases, and surjectivity in degrees $q$ and $q - 1$ makes $R^{q} f_{*} F$ finite locally free there (Cohomology and base change for proper flat coherent families, Locally free sheaves of finite rank). The unit map $O_{S} \to f_{*} O_{A}$ and its inverse by identity-section evaluation are supplied by Universal structure-sheaf sections of an abelian scheme. The finite local freeness assertion for higher direct images uses the stated successive surjectivity hypotheses.

[F4]

Every abelian variety over the fraction field of an arbitrary DVR has a finite-type Neron model by Existence of Neron models for abelian varieties over a discrete valuation ring (with AC and DC as assumed here). For such a model the local Neron-Ogg-Shafarevich criterion for $ℓ \neq char k$ : good reduction, the Neron model being an abelian scheme, inertia-fixed $ℓ$ -power torsion and an unramified Tate module are equivalent (The Neron-Ogg-Shafarevich criterion in residue characteristic prime to l, Prime-to-residue-characteristic Tate modules and inertia).

[F5]

For an abelian scheme, $[ℓ^{ν}]$ -torsion is finite etale of full rank and specializes to the geometric special torsion over a strict henselization (Abelian scheme torsion specialization is unramified, Prime to characteristic multiplication is etale).

Proof

technique · direct: read (a) and (b) off the generic fibre, Neron uniqueness and base-change stability; specialize the cohomology-and-base-change theorem for (c)-(d); and invoke the local Neron-Ogg-Shafarevich criterion for (e)

1.1F1F2givenalgebra

Clause (a): the generic fibre $A_{K}$ is an abelian variety of dimension $g$ over the function field $K$ by [F1], and the abelian scheme $A$ itself is an abelian scheme model, so $A_{K}$ has good reduction over $S$ by definition; the fibre $A_{s}$ is an abelian variety of dimension $g$ for every closed $s \in S$ by [F1]. Any abelian scheme model of $A_{K}$ is a Neron model of $A_{K}$ by [F2], and such models are uniquely isomorphic compatibly with their specified generic-fibre identifications, so $A$ has exactly this compatible uniqueness and is the Neron model.

1.2F1F2givenalgebra

Clause (b): for every $S^{'} \to S$ the base change $A_{S^{'}} \to S^{'}$ is an abelian scheme of relative dimension $g$ and represents the base change of the model by [F1]; it is therefore again a good-reduction model. If $S^{'}$ is Dedekind, $S^{'} \to S$ dominant and $K^{'}$ is its function field, then $A_{K^{'}}$ has good reduction over $S^{'}$ with model $A_{S^{'}}$ by [F2].

1.3F3givenalgebra

Clause (c): the map $φ_{s}^{q}$ is the base-change map of Cohomology and base-change map, and [F3] gives, under surjectivity, an affine open $U ∋ s$ with $h^{*} (R^{q} f_{*} F ∣_{U}) \to R^{q} f_{T *}^{'} (g_{T}^{*} F)$ an isomorphism for every $T \to U$ , and under the additional surjectivity of $φ_{s}^{q - 1}$ the finite local freeness of $R^{q} f_{*} F$ on a neighbourhood. For $q = 0$ the degree $- 1$ surjectivity condition is automatic. For $q = 0$ and $F = O_{A}$ , [F3] identifies the unit map $O_{S} \to f_{*} O_{A}$ with inverse given by evaluation along the identity section; for a line bundle flat over $S$ , the successive-surjectivity clause yields finite locally free higher direct images where the base-change maps are isomorphisms.

1.4F4F5givenalgebra

Clause (e): locally near a closed point of $S$ the base is a discrete valuation ring $R$ with fraction field $K$ and residue field $k$ , and the arbitrary-DVR existence theorem in [F4] first supplies a finite-type Neron model for the abelian variety in (e). Applying the criterion in [F4] to that model gives the equivalence between good reduction of $A_{K}$ over $R$ , the Neron model being an abelian scheme, pointwise inertia-fixed $ℓ$ -power torsion, and an unramified Tate module, for every $ℓ \neq char k$ . For an abelian scheme model, [F5] shows that each $A [ℓ^{ν}]$ is finite etale of rank $ℓ^{2 g ν}$ and commutes with every base change, specializing over a strict henselization to the special geometric torsion; this is the finite-torsion route to the criterion and claims no all-degree smooth-proper etale cohomology theorem.

2.1F2F3step 1.1step 1.2step 1.3algebra

Clause (d): the arguments in steps 1.1–1.3 are the scheme-theoretic identity, base-change and cohomology-and-base-change statements, none of which restricts the residue characteristic; only the surjectivity hypotheses of the cohomology-and-base-change theorem are used in (c), which is exactly why the finite-flatness conclusions are stated conditionally.

3.1step 1.1step 1.2step 1.3step 2.1step 1.4algebra∎

Combining steps 1.1–2.1 proves clauses (a)-(e); in particular both the coherent cohomology clauses (c)-(d) and the prime-to-residue-characteristic etale clause (e) are retained.

5 · Examples, counterexamples and false statements

None yet.