Pipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 27 results · all verified · 7 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 20 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

Chow Groups, Intersection Products, and Grothendieck-Riemann-Roch

1 · Prerequisites

2 · Summary

This page develops Chow groups and intersection theory from the cycle level up, then proves Grothendieck-Riemann-Roch for projective morphisms. It begins with algebraic cycles and the order function of a one-dimensional Noetherian local domain, defines rational equivalence and the Chow groups $A_{*}$ , and records the Grothendieck groups of coherent sheaves and of vector bundles together with their comparison on regular quasi-projective schemes. The proper pushforward and flat pullback of cycles are constructed with their norm and generic-length formulas, the compatibility of the two operations is proved for flat base change, and the localization sequence and homotopy invariance for affine spaces are established. Intersection with a Cartier divisor and the first Chern class lead to the deformation to the normal cone, the projective bundle formula, operational Chern classes with the Whitney formula, vector-bundle homotopy invariance, the Koszul and excess computations for zero sections, and the refined Gysin pullback for regular embeddings, including its commutation and composition laws. These tools build the intersection product and Chow ring of a smooth scheme and its naturality; Chern classes, the Chern character and the Todd class are then defined in the codimension-graded Chow ring with rational coefficients. The final items prove Riemann-Roch for projective-space projections and for regular embeddings, and compose the two to obtain Grothendieck-Riemann-Roch for projective morphisms of nonsingular irreducible quasi-projective varieties over an algebraically closed field, with Hirzebruch-Riemann-Roch as the point-target case. A closing remark records the conventions, the exact hypotheses, the excluded cases and the Choice assumptions of the whole chain.

3 · Logical flowchart

4 · Definitions, theorems and proofs

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Algebraic cycles and the cycle group of a scheme of finite type over a field

Definition

Let $k$ be a field and let $X$ be a scheme locally of finite type over $k$ (Locally finite type and finite type morphisms, Schemes and morphisms over a base). An algebraic cycle on $X$ is a formal $Z$ -linear combination of integral closed subschemes of $X$ (Integral schemes, Closed immersions of schemes); it is finite when the combination is finite. For an integer $d$ , the group of $d$ -cycles is the free abelian group

$Z_{d} (X) : = ⨁_{V} Z \cdot [V],$

Here the dimension of any scheme is the supremum of lengths of strict chains of nonempty irreducible closed subsets of its underlying space, with $\dim \emptyset = - \infty$ . This extends the Noetherian convention Chain dimension and the empty-space convention without requiring quasi-compactness. The direct sum runs over the integral closed subschemes $V \subseteq X$ of dimension $d$ (Chain dimension and the empty-space convention, Noetherian topological spaces via ACC on opens or DCC on closed subsets, Free abelian group on a set); for finite type $X$ one writes $Z_{*} (X) = ⨁_{d \in Z} Z_{d} (X)$ . The support $∣ α ∣$ of a cycle $α = \sum_{i} n_{i} [V_{i}]$ is the union of the $V_{i}$ with $n_{i} \neq 0$ , a closed subset of $X$ ; a cycle is effective if all $n_{i} \geq 0$ .

Conventions. (i) A cycle class may be represented by a locally finite sum in the situations below; finite sums suffice for the schemes of finite type over a field used in this pair. (ii) All integral closed subschemes are taken with the reduced structure, and $[V]$ denotes the associated basis element; the fundamental cycle of an integral $X$ of dimension $n$ is $[X] \in Z_{n} (X)$ . (iii) For $X$ equidimensional of pure dimension $n$ one writes $A^{p} (X) : = A_{n - p} (X)$ for the codimension- $p$ Chow group once $A_{*}$ is available (def-chow-group-of-cycles-mod-rational-equivalence). (iv) No choice is used in this definition beyond the free abelian group on the set of subvarieties (Free abelian group on a set). For a locally finite type scheme not assumed quasi-compact, the locally finite cycle group consists instead of formal sums whose component supports meet each quasi-compact open in only finitely many terms. Its restriction to every such open is a finite cycle. The displayed free direct sum is the finite-cycle group; throughout assertions on merely locally finite type schemes use the locally finite group. For finite type schemes the two groups coincide. For general locally finite type $X$ , $Z_{*} (X)$ denotes locally finite sums over all dimensions; it need not be the direct sum of the $Z_{d} (X)$ when component dimensions are unbounded. Every fixed-dimensional operation below is defined degreewise.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The order function of a one-dimensional Noetherian local domain

Statement

Assume the Axiom of Choice (The Axiom of Choice), used through the finite-length suppliers in (1) and the regular-local-to-DVR supplier in (4). Let $A$ be a one-dimensional Noetherian local domain with maximal ideal $m$ and fraction field $K = Frac (A)$ (A local ring is a nonzero commutative ring with a unique maximal ideal, Left and right Noetherian rings, Krull dimension of a nonzero ring, The field of fractions $Frac (D) = (D ∖ {0})^{- 1} D$ of an integral domain). For a nonzero $A$ -module $M$ of finite length write $ℓ_{A} (M)$ for its length (Composition series and length of a module). For $0 \neq r \in K$ choose $a, b \in A ∖ {0}$ with $r = a / b$ and set ${ord}_{A} (r) : = ℓ_{A} (A / a A) - ℓ_{A} (A / b A) \in Z .$ Then:

$A / a A$ and $A / b A$ have finite length, so ${ord}_{A} (r)$ is defined. For either nonzero element $x = a$ or $b$ , if $x$ is a unit then $A / x A = 0$ has length $0$ ; otherwise $A / x A$ is Noetherian and its unique prime is $m / x A$ , so it has dimension $0$ and, by the Noetherian dimension-zero finite-length theorem (using AC), finite length.
The value ${ord}_{A} (r)$ does not depend on the presentation $r = a / b$ : if $a / b = c / d$ with $a, b, c, d \neq 0$ , then $a d = b c$ , and the two exact sequences $0 \to A / a A \overset{d}{\to} A / a d A \to A / d A \to 0$ and $0 \to A / c A \overset{b}{\to} A / b c A \to A / b A \to 0$ , together with additivity of length in short exact sequences (Module length is additive in short exact sequences), give $ℓ (A / a A) - ℓ (A / b A) = ℓ (A / c A) - ℓ (A / d A)$ .
${ord}_{A} (r s) = {ord}_{A} (r) + {ord}_{A} (s)$ and ${ord}_{A} (r^{- 1}) = - {ord}_{A} (r)$ for $r, s \in K^{*}$ ; ${ord}_{A} (r) = 0$ for $r \in A^{*}$ ; ${ord}_{A} (r) \geq 0$ for $r \in A ∖ {0}$ , with equality if and only if $r \in A^{*}$ .
If $A$ is a discrete valuation ring with normalized valuation $v_{A}$ (Discrete valuation rings), then ${ord}_{A} = v_{A}$ ; if $A$ is regular of dimension one, ${ord}_{A}$ is the normalized valuation of the discrete valuation ring $A$ (Equivalent characterizations of a DVR).

Facts & Assumptions

Given: the Axiom of Choice (The Axiom of Choice); a one-dimensional Noetherian local domain $A$ with maximal ideal $m$ and fraction field $K = Frac (A)$ ; and elements $a, b, c, d \in A ∖ {0}$ together with the quotients $A / a A$ , $A / b A$ , $A / c A$ , $A / d A$ , $A / a d A = A / b c A$ and $A / r s A$ for $r, s \in A ∖ {0}$ .

[F1]

$A$ is a local ring: it has exactly one maximal ideal, namely $m$ ; it is Noetherian; it is a domain, so its zero ideal is prime and multiplication by a nonzero element of $A$ is injective; and $\dim A = 1$ , its Krull dimension as the supremum of lengths of strict chains of prime ideals (A local ring is a nonzero commutative ring with a unique maximal ideal, Left and right Noetherian rings, Zero divisor, and integral domain: a commutative ring with $1 \neq 0$ and no zero divisors, Prime ideals and maximal ideals in a commutative ring, Krull dimension of a nonzero ring). Consequently the only prime ideals of $A$ are $(0)$ and $m$ .

[F2]

For every ideal $I \subseteq A$ , contraction along the quotient map induces an inclusion-preserving bijection $Spec (A / I) \to V (I)$ from the primes of $A / I$ to the primes of $A$ containing $I$ , inverse to $p \mapsto p / I$ (Prime ideals of a quotient ring are exactly the prime ideals containing the ideal); and $A / I$ is Noetherian (Every quotient and every localisation of a Noetherian ring is Noetherian).

[F3]

Assume AC. A commutative Noetherian ring is Artinian if and only if every prime ideal is maximal (A Noetherian ring is Artinian exactly when every prime ideal is maximal); a commutative ring is Artinian if and only if its regular module has finite length (A commutative ring is Artinian exactly when it has finite length as a module over itself, Composition series and length of a module). The submodule lattice of $A / x A$ as an $A$ -module is that of the ring $A / x A$ over itself, so the two lengths agree.

[F4]

Length is additive in short exact sequences: if $0 \to N \to M \to Q \to 0$ is exact, then $M$ has finite length if and only if $N$ and $Q$ do, and then $ℓ (M) = ℓ (N) + ℓ (Q)$ (Module length is additive in short exact sequences). By Composition series and length of a module, the zero module has length $0$ and a module of finite length $0$ is zero, so a nonzero module of finite length has length at least $1$ .

[F5]

A discrete valuation ring $V$ is the valuation ring of a discrete valuation $v$ on its fraction field, and $v : K^{\times} \to Z$ is normalized to be surjective (Discrete valuation rings); a uniformizer is an element of value $1$ . A nonzero Noetherian local ring of dimension one is regular if and only if it is a discrete valuation ring, and the one-dimensional Noetherian local domain $A$ is a discrete valuation ring if and only if it is integrally closed, equivalently a local principal ideal domain with nonzero maximal ideal (one dimensional regular local rings are dvrs, Equivalent characterizations of a DVR).

Proof

technique · direct; compute the two exact sequences and read off well-definedness, multiplicativity and the valuation comparison

1.1F1F2F3given

Finiteness and the cases $x$ unit or nonunit. Let $x \in A$ be nonzero. If $x$ is a unit, then $A / x A = 0$ , its spectrum is empty, and its length as an $A$ -module is $0$ . If $x$ is a nonunit, then $x A ⊊ A$ and $x \in m$ . A prime of $A / x A$ corresponds by [F2] to a prime of $A$ containing $x$ ; the primes of $A$ are $(0)$ and $m$ by [F1], and $x \notin (0)$ because $x \neq 0$ while $A$ is a domain. Thus the only prime of $A / x A$ is $m / x A$ , which is proper and maximal since $x A \subseteq m$ . By [F2], $A / x A$ is Noetherian; every one of its prime ideals is maximal, so it is Artinian and its regular module has finite length by [F3]. Applying these two cases to $x = a, b, c, d$ and $x y$ for nonzero $x, y$ shows that $ℓ_{A} (A / a A)$ and $ℓ_{A} (A / b A)$ , and likewise $ℓ_{A} (A / c A)$ , $ℓ_{A} (A / d A)$ and $ℓ_{A} (A / x y A)$ , are defined natural numbers.

2.1F1F4step 1.1algebra

Additivity of the length of $A / x y A$ . For $x, y \neq 0$ the sequence $0 ⟶ A / x A \overset{y}{\to} A / x y A ⟶ A / y A ⟶ 0$ is exact: multiplication by $y$ is well defined because $y (x) \subseteq (x y)$ , it is injective because $y$ is a nonzerodivisor of the domain $A$ , its image is the submodule $(y) / (x y)$ of $A / x y A$ , and the reduction $A / x y A \to A / y A$ is well defined with kernel exactly $(y) / (x y)$ . Since all three modules have finite length by step 1.1, [F4] gives $ℓ_{A} (A / x y A) = ℓ_{A} (A / x A) + ℓ_{A} (A / y A)$ .

2.2F4step 1.1algebra

Values on $A$ and units. If $r = u \in A^{\times}$ , then ${ord}_{A} (u) = ℓ_{A} (A / u A) - ℓ_{A} (A / A) = ℓ_{A} (0) - 0 = 0$ by [F4]. If $r = x \in A ∖ {0}$ , then ${ord}_{A} (x) = ℓ_{A} (A / x A) - ℓ_{A} (A / A) = ℓ_{A} (A / x A) \geq 0$ by [F4], and equality holds if and only if $ℓ_{A} (A / x A) = 0$ , if and only if $A / x A = 0$ by [F4], if and only if $x$ is a unit of $A$ .

3.1step 2.1algebra

Independence of the presentation. Suppose $a / b = c / d$ with $a, b, c, d \neq 0$ ; then $a d = b c$ by the usual fraction comparison in the domain $A$ . The sequence $0 \to A / a A \overset{d}{\to} A / a d A \to A / d A \to 0$ is exact by the argument of step 2.1, so by additivity $ℓ_{A} (A / a d A) = ℓ_{A} (A / a A) + ℓ_{A} (A / d A)$ ; likewise the sequence $0 \to A / c A \overset{b}{\to} A / b c A \to A / b A \to 0$ gives $ℓ_{A} (A / b c A) = ℓ_{A} (A / c A) + ℓ_{A} (A / b A)$ . Since $A / a d A = A / b c A$ as $a d = b c$ , subtracting the two identities yields $ℓ_{A} (A / a A) - ℓ_{A} (A / b A) = ℓ_{A} (A / c A) - ℓ_{A} (A / d A)$ . Hence ${ord}_{A} (r)$ is independent of the chosen presentation of $r$ .

3.2step 2.1algebra

Multiplicativity and inverse. Let $r = a / b$ and $s = c / d$ with $a, b, c, d \neq 0$ , so that $r s = a c / b d$ . By step 2.1, $ℓ_{A} (A / a c A) = ℓ_{A} (A / a A) + ℓ_{A} (A / c A)$ and $ℓ_{A} (A / b d A) = ℓ_{A} (A / b A) + ℓ_{A} (A / d A)$ . Therefore ${ord}_{A} (r s) = ℓ_{A} (A / a c A) - ℓ_{A} (A / b d A) = (ℓ_{A} (A / a A) - ℓ_{A} (A / b A)) + (ℓ_{A} (A / c A) - ℓ_{A} (A / d A)) = {ord}_{A} (r) + {ord}_{A} (s) .$ Applying this with $r s = 1$ and taking $r = a / b$ , $s = b / a$ gives ${ord}_{A} (r^{- 1}) = {ord}_{A} (b / a) = - {ord}_{A} (a / b) = - {ord}_{A} (r)$ .

4.1F5step 1.1algebra∎

The discrete valuation ring case. Suppose $A$ is a discrete valuation ring with normalized valuation $v_{A}$ (Discrete valuation rings). Choose a uniformizer $π$ of $A$ , an element with $v_{A} (π) = 1$ ; it exists because $v_{A}$ is normalized. Every $0 \neq x \in A$ has $v_{A} (x) = n \geq 0$ and can be written $x = u π^{n}$ with $u \in A^{\times}$ , because $x π^{- n}$ has value $0$ and is therefore a unit of $A$ . Multiplication by $u$ is an automorphism of $A$ carrying $(x)$ to $(π^{n})$ , so $A / x A ≅ A / π^{n} A$ , and the chain $A ⊋ (π) ⊋ \dots ⊋ (π^{n})$ exhibits $A / π^{n} A$ as an iterated extension of the field $A / π A$ , of length $n$ : indeed each successive quotient $(π^{i}) / (π^{i + 1})$ is isomorphic to $A / π A$ , a field, hence simple. Thus $ℓ_{A} (A / x A) = n = v_{A} (x)$ for every $0 \neq x \in A$ , and for $r = a / b$ with $a, b \neq 0$ we get ${ord}_{A} (r) = v_{A} (a) - v_{A} (b) = v_{A} (r)$ , since $v_{A}$ is a group homomorphism $K^{\times} \to Z$ and $v_{A} (b^{- 1}) = - v_{A} (b)$ . If $A$ is regular of dimension one, then the same argument applies because $A$ is a discrete valuation ring by [F5].

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

Grothendieck groups of coherent sheaves and of vector bundles on a scheme

Definition

Assume the Axiom of Choice (The Axiom of Choice) inherited from the coherent-category and higher-direct-image suppliers. Let $X$ be a locally Noetherian scheme (Locally Noetherian and Noetherian schemes). Then the category of coherent $O_{X}$ -modules is abelian (Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves), and the category of finite locally free $O_{X}$ -modules of locally constant rank is an exact category (Locally free sheaves of finite rank).

The Grothendieck group of coherent sheaves $K_{0} (X) : = K_{0} (Coh (X))$ is the abelian group with one generator $[F]$ for each isomorphism class of coherent sheaves, subject to $[F] = [F^{'}] + [F^{''}]$ for every short exact sequence $0 \to F^{'} \to F \to F^{''} \to 0$ (Grothendieck group of an essentially small abelian category); existence of the group is the group completion of the commutative monoid modulo the exact-sequence relations. These isomorphism classes form a set: on a fixed set-indexed affine cover, finite presentations give a set of local module models, and their overlap isomorphisms form sets; gluing those data gives a set of representatives. The same applies to vector bundles, using finite free local models.
The Grothendieck group of vector bundles $K^{0} (X)$ is the abelian group with one generator $[E]$ for each isomorphism class of finite locally free sheaves of locally constant rank, subject to $[E] = [E^{'}] + [E^{''}]$ for every exact sequence $0 \to E^{'} \to E \to E^{''} \to 0$ of such sheaves. Tensor product makes $K^{0} (X)$ a commutative ring with unit $[O_{X}]$ (Tensor product of sheaves of modules, Tensor product preserves quasi-coherence), and makes $K_{0} (X)$ a $K^{0} (X)$ -module via $[E] \cdot [F] : = [E \otimes F]$ , using that tensoring with a locally free sheaf is exact.
Every morphism $f : X \to Y$ of locally Noetherian schemes induces $f^{*} : K^{0} (Y) \to K^{0} (X)$ by pullback of locally free sheaves (Scheme pullback preserves quasi-coherence, Dual and base change for finite locally free sheaves); flat $f$ also induces $f^{*} : K_{0} (Y) \to K_{0} (X)$ .
Under the Axiom of Choice (The Axiom of Choice) inherited from the coherent-cohomology suppliers, if $f : X \to Y$ is proper between finite type schemes over a field, the higher direct images $R^{q} f_{*} F$ are coherent (Coherent higher direct images under proper morphisms) and vanish for $q ≫ 0$ over affine opens of $Y$ (Dimension bound for quasi-coherent cohomology on a Noetherian scheme, with the global bound $q > \dim X$ ); the resulting pushforward $f_{!} : K_{0} (X) \to K_{0} (Y)$ is defined in def-pushforward-in-algebraic-k-theory.
The natural map $K^{0} (X) \to K_{0} (X)$ , $[E] \mapsto [E]$ , is well defined and additive; under the same Axiom of Choice premise it is an isomorphism when $X$ is a regular quasi-projective scheme of finite type over a field (lem-k-zero-vector-bundles-versus-coherent-sheaves).

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Smooth immersions, their conormal sequence, deformation charts and smooth sections

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the regular-local and smooth differential suppliers. For a closed immersion $i : X ↪ Y$ of smooth finite type schemes over a field, with pure dimensions $n, m$ and constant codimension $d = m - n$ , the ideal is locally generated by a regular sequence of length $d$ , $I / I^{2}$ is locally free of rank $d$ , and $0 \to I / I^{2} \to i^{*} Ω_{Y / k}^{1} \to Ω_{X / k}^{1} \to 0$ is exact. Its dual gives $0 \to T_{X} \to i^{*} T_{Y} \to N_{X / Y} \to 0$ . Locally in the étale topology the embedding is the inverse image of the coordinate inclusion $A^{n} \subset A^{m}$ . Consequently ${Bl}_{X \times {\infty}} (Y \times P^{1})$ is smooth; if $Y$ is quasi-projective, this blowup is quasi-projective. The same immersion and normal assertions hold on locally closed presentations, by restriction. More generally, a section $s : T \to P$ of a smooth morphism of locally Noetherian schemes of constant relative dimension $r$ is a regular immersion of codimension $r$ . Its regular equations and normal bundle commute with every base change on $T$ , including singular bases.

Facts & Assumptions

Given: the Axiom of Choice; a field $k$ ; a closed immersion $i : X ↪ Y$ of smooth finite type $k$ -schemes with $X$ of pure dimension $n$ , $Y$ of pure dimension $m$ and constant codimension $d = m - n$ ; its ideal sheaf $I \subseteq O_{Y}$ ; and, in the last part, a section $s : T \to P$ of a smooth morphism $P \to T$ of locally Noetherian schemes of constant relative dimension $r$ .

[F1]

Jacobian criterion. A morphism $f$ locally of finite presentation is smooth at $x$ if and only if near $x$ it admits a standard smooth chart: affine opens $Spec C$ of $x$ and $Spec A$ of $f (x)$ with $f (Spec C) \subseteq Spec A$ and an isomorphism $C_{h} ≅ (A [t_{1}, \dots, t_{N}] / (g_{1}, \dots, g_{r}))_{β}$ in which some $r \times r$ minor of the Jacobian matrix $(\partial g_{j} / \partial t_{i})$ is a unit; such a chart is flat over $A$ , exhibits relative dimension $N - r$ at $x$ , and its fibres are geometrically regular (Relative Jacobian criterion with its presentation hypothesis, Smooth morphism of schemes, Relative dimension of a smooth morphism at a point).

[F2]

Smoothness of a finite type $k$ -algebra and regularity. A finite type $k$ -algebra admitting a standard smooth presentation at every prime is geometrically regular over $k$ (Locally standard smooth iff flat with geometrically regular fibres, Standard smooth presentations and locally standard smooth maps, Geometrically regular algebras and geometrically regular fibres); geometric regularity over $k$ implies that every local ring is regular local, since the empty generating list exhibits $k$ itself as a finitely generated field extension of $k$ . At a prime admitting a standard smooth chart, the local ring is the localization of that chart and is regular. Moreover every local ring of a standard smooth $k$ -algebra at a prime $Q$ is regular of dimension $ht (Q^{'}) - c$ , where $Q^{'}$ is the corresponding prime of the polynomial ring and $c$ the number of equations (Fibres of standard smooth algebras are regular of relative dimension). A field is Noetherian, and a finite type algebra over a Noetherian ring is Noetherian (A field has only the zero ideal and itself, hence is Noetherian, Every algebra of finite type over a Noetherian ring is a Noetherian ring).

[F3]

Local dimension formula. For a finite type $k$ -scheme $Z$ , a point $z$ and the infimum $\dim_{z} Z$ of the dimensions of open neighbourhoods of $z$ , one has $\dim_{z} Z = \dim O_{Z, z} + {trdeg}_{k} κ (z)$ (Local fibre dimension equals local ring dimension plus residue transcendence degree). A smooth pure-dimensional $k$ -scheme of dimension $e$ has local scheme dimension $e$ at every point.

[F4]

Regular quotients. If $(R, m)$ is regular local and $I \subseteq m$ is an ideal with $R / I$ regular, then $I$ is generated by an initial segment of a regular system of parameters, of length $\dim R - \dim (R / I)$ , so its members form an $R$ -regular sequence (regular local regular quotient ideal is parameter generated).

[F5]

Regular sequences and Koszul. Every finite regular sequence is Koszul-regular: its positive-degree Koszul homology vanishes, so every relation $\sum_{j} a_{j} f_{j} = 0$ among the sequence elements is a combination of the Koszul relations $f_{j} e_{k} - f_{k} e_{j}$ (Regular Sequences Give Acyclic Koszul Complexes).

[F6]

Conormal sequence. For a closed immersion $i : X \to Y$ of $S$ -schemes with ideal sheaf $I$ , the sequence $I / I^{2} \to i^{*} Ω_{Y / S}^{1} \to Ω_{X / S}^{1} \to 0$ of $O_{X}$ -modules is exact (Conormal sequence for a closed immersion).

[F7]

Differential sheaves. A morphism smooth at a point has locally free relative differentials of rank equal to the relative dimension there (Differentials of a smooth morphism, Relative dimension of a smooth morphism at a point). A morphism is formally unramified if and only if its relative differentials vanish; unramified means locally of finite type plus formally unramified; and étale means flat plus unramified for a locally finitely presented morphism (Formal unramifiedness iff Omega vanishes, Étale morphism of schemes, Étale equals flat and unramified in finite presentation). Smooth and étale morphisms are stable under base change and composition (Smoothness survives base change and composition, Étale stability).

[F8]

Smooth local structure. A smooth morphism of relative dimension $r$ at a point is, after shrinking the source and the base Zariski locally, the composite of an étale morphism to $A_{T}^{r}$ and the projection (Smooth maps have étale local affine-space form, Relative dimension of a smooth morphism at a point). An étale morphism is locally standard étale: $B ≅ (A [z] / (P))_{g}$ with $P$ monic and $P^{'}$ a unit (Étale morphisms are locally standard étale).

[F9]

Blowups. Blowing up a finite type quasi-coherent ideal sheaf is locally H-projective; if the ideal is globally generated by $f_{0}, \dots, f_{r}$ the blowup embeds as a closed subscheme of $P_{X}^{r}$ (Blowups of finite type ideals are locally H-projective, and proper, Blowup of a scheme along an ideal sheaf, Rees algebra sheaf of a finite type ideal). Blowups commute with flat base change (Flat base change for blowups, and failure without flatness).

[F10]

Ampleness. On a quasi-projective $k$ -scheme a very ample invertible sheaf exists, hence an ample one (Relative very ampleness implies relative ampleness, Quasi-projective morphisms before Proj); for an ample invertible sheaf $L$ and a coherent sheaf $F$ , the twist $F \otimes L^{\otimes ν}$ is globally generated for all $ν ≫ 0$ (Serre global-generation criterion for ampleness). The Segre embedding exhibits $P_{k}^{m_{1}} \times_{k} P_{k}^{m_{2}}$ as a closed subscheme of a projective space, so a product of quasi-projective $k$ -schemes is quasi-projective when the embedding is quasi-compact (The Segre-Veronese map is a closed embedding).

Proof

technique · direct; reduce the immersion to its local regular-sequence model, compute the conormal sequence and its dual, exhibit the étale-local coordinate model, and read the deformation and section statements off that model

1.1F1F2F3given

Local invariants at a point. Fix $x \in X$ and put $R = O_{Y, x}$ , $S = O_{X, x}$ , so that the closed immersion identifies $S$ with $R / I_{x}$ and $κ (x)$ with the residue field of both. Since $Y$ and $X$ are smooth of finite type over $k$ at $x$ , [F1] provides standard smooth charts through $x$ and through $x$ in $X$ ; by [F2] the local rings $R$ and $S$ are regular local rings. Write $t = {trdeg}_{k} κ (x)$ . Both $Y$ and $X$ are pure-dimensional and smooth, so their local scheme dimensions at $x$ equal $m$ and $n$ ; by [F3] applied to each, $\dim R = m - t$ and $\dim S = n - t$ , hence $\dim R - \dim S = m - n = d$ .

2.1F2F4step 1.1algebra

A regular sequence locally generating the ideal. By step 1.1 the local ring $R$ is regular and its quotient $S = R / I_{x}$ is regular, so [F4] applied to $I = I_{x}$ produces a regular system of parameters of $R$ whose first $d$ members $f_{1}, \dots, f_{d}$ generate $I_{x}$ ; in particular $f_{1}, \dots, f_{d}$ is an $R$ -regular sequence. Each germ has a representative on a common affine neighbourhood $U$ of $x$ . That neighbourhood is Noetherian by [F2], so its ideal of definition of $U \cap X$ is finitely generated, and clearing denominators in the finitely many generation equalities shrinks $U$ until the representatives generate the ideal sheaf and their germs at $x$ remain the same regular sequence.

3.1F5step 2.1algebra

The conormal sheaf is free with basis the classes of the $f_{j}$ . Let $S^{d} \to I / I^{2}$ be the surjection sending the $j$ -th basis vector to the class of $f_{j}$ , which is onto by step 2.1. If $\sum_{j} {\overset{ˉ}{a}}_{j} [f_{j}] = 0$ in $I / I^{2}$ , lift the ${\overset{ˉ}{a}}_{j}$ to $a_{j} \in R$ ; the relation says $\sum_{j} a_{j} f_{j} \in I^{2}$ , say $\sum_{j} a_{j} f_{j} = \sum_{j} b_{j} f_{j}$ with all $b_{j} \in I$ , so $\sum_{j} (a_{j} - b_{j}) f_{j} = 0$ . By [F5] every $a_{j} - b_{j}$ lies in $I$ , whence every ${\overset{ˉ}{a}}_{j} = 0$ . Thus $S^{d} \to I / I^{2}$ is an isomorphism, and since $x$ was arbitrary $I / I^{2}$ is locally free of rank $d$ with local basis the classes of $f_{1}, \dots, f_{d}$ .

4.1F6F7step 3.1algebra

Exactness of the conormal sequence and its dual. By [F6] the conormal map $I / I^{2} \to i^{*} Ω_{Y / k}^{1}$ surjects onto the kernel $K$ of $i^{*} Ω_{Y / k}^{1} \to Ω_{X / k}^{1}$ . At $x$ , [F7] makes $i^{*} Ω_{Y / k, x}^{1}$ and $Ω_{X / k, x}^{1}$ free over $S$ of ranks $m$ and $n$ ; the surjection onto the free target splits, so $K$ is free of rank $m - n = d$ . By step 3.1 the conormal module is free of rank $d$ , so the surjection between free modules of equal rank has nonzero determinant modulo the maximal ideal, hence is an isomorphism. Therefore the conormal sequence is short exact. Dualizing it at each point, the dual of a short exact sequence of finite free modules over a local ring is short exact, so $0 \to T_{X} \to i^{*} T_{Y} \to N_{X / Y} \to 0$ is exact with $N_{X / Y} = (I / I^{2})^{\lor}$ locally free of rank $d$ .

5.1F1F7step 2.1step 4.1algebra

The étale-local coordinate model. Shrinking around $x$ , choose $u_{1}, \dots, u_{n} \in O_{Y} (U)$ whose differentials on $X$ form a basis of $Ω_{X / k}^{1}$ near $x$ : differentials of local functions generate this sheaf, so select $n$ of them whose residues form a basis at $x$ , then shrink until their determinant is a unit. Lift these functions through the surjection $O_{Y} (U) \to O_{X} (U \cap X)$ . Form the $k$ -morphism $φ = (φ_{1}, \dots, φ_{m}) : U \to A_{k}^{m}$ with coordinates $φ_{j} = f_{j}$ for $j \leq d$ and $φ_{d + j} = u_{j}$ . By step 4.1 the classes of $d f_{1}, \dots, d f_{d}$ form a basis of $I / I^{2} \otimes κ (x)$ and the classes of $d u_{1}, \dots, d u_{n}$ a basis of $Ω_{X / k}^{1} \otimes κ (x)$ ; the exact conormal sequence of step 4.1 then shows that $d φ_{1}, \dots, d φ_{m}$ span the $m$ -dimensional $κ (x)$ -vector space $Ω_{Y / k, x}^{1} \otimes κ (x)$ . Choosing a presentation of $U$ as an étale-locally standard smooth chart as in [F1] and adjoining the $m$ equations $φ_{j} - t_{j}$ exhibits a standard smooth presentation of $U \to A_{k}^{m}$ at $x$ whose Jacobian matrix has an invertible full $m \times m$ minor, because the corresponding row space together with the rows of the chart presentation spans. Hence $φ$ is smooth of relative dimension $0$ at $x$ by [F1]. Its relative differentials vanish at $x$ by [F7], so $φ$ is unramified and, being flat and locally of finite presentation by [F1], étale at $x$ by [F7]. Finally, $φ^{- 1} (A^{n}) \cap U$ , where $A^{n} \subset A^{m}$ is the coordinate subspace $t_{1} = \dots = t_{d} = 0$ , is exactly $U \cap X$ , because $X$ is locally defined in $U$ by $f_{1} = \dots = f_{d} = 0$ by step 2.1.

5.2F6step 4.1algebra

The locally closed case. If $i : X ↪ Y$ is only a locally closed immersion, write it as a closed immersion $X ↪ U$ followed by an open immersion $U ↪ Y$ ; the ideal sheaf, the conormal sheaf and the conormal sequence are those of the closed part, computed in $U$ , and the assertions are local near points of $X$ , so they follow from the closed case applied to $X ↪ U$ .

6.1F1F9step 5.1algebra

Smoothness of the deformation blowup. On the affine chart of $P^{1}$ containing $\infty$ , use the coordinate $t$ vanishing at $\infty$ ; the centre is cut out by $(f_{1}, \dots, f_{d}, t)$ . On its complement the centre is empty, so the blowup is the identity and is smooth. Near the centre the coordinate model is the blowup of $A^{m} \times A^{1}$ along $(f_{1}, \dots, f_{d}, t)$ . Its charts are computed from the Rees algebra: the $t$ -chart has coordinates $(f_{1} / t, \dots, f_{d} / t, t, u_{1}, \dots, u_{n})$ and the $f_{j}$ -charts have coordinates $(f_{j}, f_{l} / f_{j} (l \neq j), t / f_{j}, u_{1}, \dots, u_{n})$ , each an affine space of dimension $m + 1$ ; the chart ideals are generated by regular sequences, so each chart is smooth over $k$ , and smoothness is local on the source, so the model blowup is smooth. Now let $ψ = φ \times {id}_{P^{1}} : U \times P^{1} \to A^{m} \times P^{1}$ . It is étale, hence flat, and the ideal of $X \times {\infty}$ in $U \times P^{1}$ is the inverse image of the model ideal because $X \cap U = φ^{- 1} (A^{n})$ . By [F9] (flat base change of blowups) the blowup of $U \times P^{1}$ along $X \times {\infty}$ is the base change of the smooth model blowup along $ψ$ ; by [F7] base changes of smooth morphisms are smooth, and a composition of smooth morphisms is smooth, so this base change is smooth over $k$ . As $x \in X$ was arbitrary and $X$ is covered by the opens $U \cap X$ , the blowup ${Bl}_{X \times {\infty}} (Y \times P^{1})$ is smooth over $k$ .

7.1F9F10step 6.1algebra

Quasi-projectivity of the blowup. Assume now that $Y$ is quasi-projective over $k$ . Then $Y \times P_{k}^{1}$ is quasi-projective: $Y$ is a quasi-compact locally closed subscheme of some projective space, and the Segre embedding [F10] realizes the product of projective spaces as a closed subscheme of a projective space, so the product immerses quasi-compactly. Let $Z = Y \times P_{k}^{1}$ and let $L$ be very ample on $Z$ , hence ample [F10]. The ideal sheaf $J$ of $X \times {\infty}$ is coherent, so by [F10] the twist $J \otimes L^{\otimes ν}$ is globally generated for some $ν \geq 1$ ; the graded algebra $⨁_{q \geq 0} J^{q} \otimes L^{ν q}$ has the same relative Proj as $⨁_{q \geq 0} J^{q}$ , since a trivialization of $L$ identifies their Proj charts and changing the trivialization rescales degree $q$ by the $q$ -th power of a unit. A finite set of generators of $J \otimes L^{ν}$ therefore gives a graded surjection from $O_{Z} [T_{0}, \dots, T_{r}]$ onto this twisted Rees algebra. Its relative Proj is a closed subscheme of $P_{Z}^{r}$ , by the same homogeneous-quotient chart calculation as [F9], and a projective bundle over the quasi-projective $Z$ is quasi-projective, so the blowup is quasi-projective over $k$ .

8.1F7F8algebra∎

Sections of smooth morphisms. Let $s : T \to P$ be a section of a smooth morphism $P \to T$ of constant relative dimension $r$ , with $T$ locally Noetherian. Near a point $t \in T$ , [F8] factors $P \to T$ Zariski locally as an étale morphism $P^{'} \to A_{T}^{r} = Spec O_{T} [x_{1}, \dots, x_{r}]$ followed by the projection, after shrinking around $s (t)$ and $t$ ; by [F7] the étale morphism is flat. The composite $T \to P^{'} \to A_{T}^{r}$ is the graph of the $T$ -point with coordinates $c_{i} = s^{*} (x_{i})$ . This graph is cut out by the regular sequence $x_{1} - c_{1}, \dots, x_{r} - c_{r}$ : after quotienting by the first $i$ terms, the next element is a monic linear polynomial in a new variable and hence a nonzerodivisor over any base ring. Pulling the graph back along the flat étale map $P^{'} \to A_{T}^{r}$ gives a regular immersion $Q = T \times_{A_{T}^{r}} P^{'} ↪ P^{'}$ of codimension $r$ , and $Q \to T$ is étale. The section factors through a section $T \to Q$ of this étale morphism. By [F8], after shrinking near the chosen point, this section is described by a standard étale chart $Spec ((A [z] / (P))_{g}) \to Spec A$ with $z \mapsto a$ , $P (a) = 0$ and $P^{'} (a)$ a unit. Factoring $P (z) = (z - a) Q_{0} (z)$ gives $Q_{0} (a) = P^{'} (a)$ ; after restricting to the open neighborhood where $Q_{0}$ is invertible, the equation $P (z) = 0$ forces $z = a$ , so the section is an open immersion there. Hence near its image in $P^{'}$ , the pulled-back regular equations cut out $s (T)$ , proving that $s$ is a regular immersion of codimension $r$ . These equations remain regular after every base change $T^{'} \to T$ , since the section of the smooth base change $P \times_{T} T^{'} \to T^{'}$ is again locally obtained by this graph construction, including when $T^{'}$ is singular. The normal bundle also commutes with base change: for a section of a smooth morphism, the conormal bundle identifies with $s^{*} Ω_{P / T}^{1}$ , and this relative differential bundle pulls back to $s^{' *} Ω_{(P \times_{T} T^{'}) / T^{'}}^{1}$ . Hence the codimension and normal bundle are preserved by arbitrary base change on $T$ .

DefinitionDefinition: AI-adaptedProof: Not applicableOpen item page →

Rational equivalence and the Chow group of cycles

Definition

Assume the Axiom of Choice (The Axiom of Choice), inherited from the order function of The order function of a one-dimensional Noetherian local domain; it is used there for the finiteness of the lengths, and here only through that order function. Let $k$ be a field and let $X$ be a scheme locally of finite type over $k$ . For $d \in Z$ let ${Rat}_{d} (X) \subseteq Z_{d} (X)$ be the subgroup generated by all cycles of the form $(i_{W})_{*} div (r), div (r) = \sum_{V \subseteq W} {ord}_{O_{W, V}} (r) [V],$ where $W \subseteq X$ is an integral closed subscheme of dimension $d + 1$ (Integral schemes), $i_{W} : W ↪ X$ is the inclusion, $r \in k (W)^{*}$ is a nonzero rational function on $W$ (Sheaf total quotient rings, The field of fractions $Frac (D) = (D ∖ {0})^{- 1} D$ of an integral domain), the sum runs over the integral closed subschemes $V \subseteq W$ of dimension $d$ , and ${ord}_{O_{W, V}} (r)$ is the order function of The order function of a one-dimensional Noetherian local domain for the one-dimensional Noetherian local domain $O_{W, V}$ (A local ring is a nonzero commutative ring with a unique maximal ideal). The sum is locally finite, and finite when $W$ is of finite type, as proved below. The elements of ${Rat}_{d} (X)$ are the $d$ -cycles rationally equivalent to zero; two cycles are rationally equivalent if their difference is, written $α \sim_{r a t} β$ . The Chow group of $d$ -cycles is $A_{d} (X) : = Z_{d} (X) / {Rat}_{d} (X), A_{*} (X) = ⨁_{d} A_{d} (X) for finite type X .$

Basic facts. (i) Rational equivalence is an equivalence relation compatible with the group structure, by definition. (ii) If $W$ is a normal integral closed subscheme, then $div (r)$ agrees with the principal Weil divisor of $r$ computed with the divisor theory of Order codimension one rational function, Principal weil divisor and class group; in particular for $X$ integral of dimension $n$ and normal (e.g. smooth), $A_{n} (X) ≅ Z$ generated by $[X]$ and $A_{d} (X) = 0$ for $d > \dim X$ . (iii) For $X$ equidimensional of pure dimension $n$ , set $A^{p} (X) : = A_{n - p} (X)$ ; this is the codimension grading used throughout. (iv) Equivalent presentation: $α \sim_{r a t} 0$ if and only if $α = \sum_{i} (i_{W_{i}})_{*} div (r_{i})$ as above; this is the "Fulton 1.6" definition with supports on closed subschemes of $X \times P^{1}$ specialized, and it is Stacks' Definition 42.19.1 (tag 02RW). For merely locally finite type $X$ , use the locally finite cycle group of Algebraic cycles and the cycle group of a scheme of finite type over a field and locally finite sums of principal divisors supported on locally finite families of the subvarieties $W$ . These sums define the rational-equivalence subgroup; finite generation as written above applies when $X$ is of finite type. In the general case the total group $A_{*} (X)$ is the quotient of the total locally finite cycle group by locally finite sums of these relations, allowing unbounded dimensions as in Algebraic cycles and the cycle group of a scheme of finite type over a field. Thus pullback along a non-quasi-compact flat map uses locally finite cycles, not a finite-support sum with infinitely many components.

Well-definedness of the divisor sum. Let $W$ be integral of dimension $d + 1$ and $0 \neq r \in k (W)^{*}$ . On an affine open chart $Spec A \subseteq W$ with $W$ of finite type over $k$ , write $r = a / b$ with $a, b \in A ∖ {0}$ under the identification $k (W) = Frac (A)$ . For an integral closed subscheme $V \subseteq W$ of dimension $d$ meeting the chart in a height-one prime $p$ of $A$ , the order ${ord}_{O_{W, V}} (r)$ of The order function of a one-dimensional Noetherian local domain vanishes unless $p$ contains $a$ or $b$ , hence unless $p$ contains $a b$ ; then $p$ is minimal over the principal ideal $(a b)$ , hence has height one by A minimal prime over a principal nonzerodivisor has height one. A Noetherian ring has only finitely many minimal primes over the radical of a principal ideal (A radical ideal in a Noetherian ring is a finite intersection of minimal primes), so only finitely many $V$ meet a fixed affine chart with nonzero coefficient; since $W$ of finite type over a field is quasi-compact, a finite affine cover exhibits $div (r)$ as a finite sum, and for merely locally finite type $W$ it is a locally finite sum in the sense of Algebraic cycles and the cycle group of a scheme of finite type over a field. Thus the displayed sum defines an element of $Z_{d} (W)$ in the finite-type case and of the locally finite cycle group in general, and the subgroup ${Rat}_{d} (X)$ is well defined.

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

Cycles of coherent sheaves and of closed subschemes, with flat pullback

Statement

Assume the Axiom of Choice (The Axiom of Choice) for the going-down prime-lifting supplier used in flat pullback. Let $X$ be a scheme locally of finite type over a field; use finite cycles if $X$ is of finite type, and locally finite cycles otherwise. For each integer $d$ , a coherent sheaf $F$ whose support has dimension at most $d$ has the $d$ -cycle $[F]_{d} = \sum_{\dim V = d} ℓ_{O_{X, η_{V}}} (F_{η_{V}}) [V],$ where $V$ runs over the integral closed subschemes and $η_{V}$ is their generic point. The sum is locally finite, and finite for finite type $X$ ; its terms are precisely the dimension- $d$ components of the support. In any short exact sequence of coherent sheaves all supported in dimension at most $d$ , these $d$ -cycles are additive. No additivity is claimed for the sum of cycles in all dimensions when the supports change.

For a closed subscheme $Y$ , define its fundamental cycle $[Y]$ by summing the generic lengths at every irreducible component of $Y$ . If $Y$ is pure dimensional of dimension $d$ , $[Y] = [O_{Y}]_{d}$ . For reduced $Y$ every component coefficient is one.

For $X^{'}$ locally of finite type over the same field and flat $f : X^{'} \to X$ with every nonempty fibre pure of dimension $r$ , $[f^{- 1} Y]_{d + r} = f^{*} [Y]_{d}$ when $Y$ is pure of dimension $d$ . If $Y$ is pure of dimension $d$ and the local equation of an effective Cartier divisor $D$ of $X$ is a nonzerodivisor on $O_{Y}$ , then $D ∣_{Y}$ is an effective Cartier divisor and its cycle is the fundamental $(d - 1)$ -cycle of $Y \cap D$ . The nonzerodivisor condition cannot be replaced by $Y \subset̸ D$ : it must exclude every associated point of $Y$ , including embedded ones.

Facts & Assumptions

Given: the Axiom of Choice; a scheme $X$ locally of finite type over a field; a coherent $O_{X}$ -module $F$ with support of dimension at most $d$ ; a closed subscheme $Y \subseteq X$ ; a scheme $X^{'}$ locally of finite type over the same field and a flat morphism $f : X^{'} \to X$ with every nonempty fibre pure of dimension $r$ ; an effective Cartier divisor $D \subseteq X$ .

[L1]

Cycles are formal $Z$ -linear combinations of integral closed subschemes, with finite support in the finite type case and locally finite support in general (Algebraic cycles and the cycle group of a scheme of finite type over a field, Generic points of irreducible closed subsets).

[L2]

On a locally Noetherian scheme, coherence is a local condition of finite presentation; the support of a coherent sheaf is closed, and the local ring of a reduced scheme at the generic point of an irreducible component is a field, its function field (Coherent module sheaves, A local ring is a nonzero commutative ring with a unique maximal ideal, Sheaf total quotient rings, Irreducible components as schemes).

[L3]

A nonzero finitely generated module over a field has a composition series, hence finite length, and length is additive in short exact sequences (Composition series and length of a module, Module length is additive in short exact sequences).

[L4]

A sheaf whose stalks vanish off a closed subset is the pushforward of its restriction to that subset (A sheaf with no stalks off a closed subset is a pushforward). For a coherent module at a generic support point, finite length is established directly by the annihilator filtration in step 1.1, using the composition-series definition (Composition series and length of a module).

[L5]

An effective Cartier divisor on a locally Noetherian scheme is locally defined by a nonzerodivisor, and its restriction to a closed subscheme on which the local equation remains a nonzerodivisor is again an effective Cartier divisor (Closed immersions of schemes, A local ring is a nonzero commutative ring with a unique maximal ideal). Flatness means flatness of the stalk maps (Flat morphism of schemes); pure dimension $r$ of the nonempty fibres is a separate hypothesis. Components of a flat preimage dominate components of its base by going down; the dimension formula then gives the shift by $r$ (Every flat ring map satisfies going down, Affine-domain dimension equals transcendence degree, Transcendence degree is additive in finite towers).

Proof

technique · direct; compute each coefficient after localizing at the generic points of the relevant components

1.1L1L2L3L4given

Finiteness of the coefficient sum. Let $V$ be a $d$ -dimensional integral closed subscheme of $X$ whose generic point lies in the support of $F$ . Put $R = O_{X, η_{V}}$ and $M = F_{η_{V}}$ . The dimension bound makes $V$ a component of the support whenever $M \neq 0$ : a strictly larger integral subvariety has strictly larger dimension on a finite type affine neighbourhood. Thus $M$ is finitely generated and its support in $Spec R$ is only the maximal ideal $m$ . Consequently $\sqrt{{Ann}_{R} (M)} = m$ . Choose finite generators of $m$ ; a power of each lies in the annihilator, so a sufficiently large power $m^{N}$ annihilates $M$ . The finite filtration $M \supseteq m M \supseteq \dots \supseteq m^{N} M = 0$ has finite-dimensional residue-field quotients. Refining their finite vector-space filtrations proves finite length over $R$ , without asserting that $R$ itself is a field. Hence every coefficient $ℓ_{O_{X, η_{V}}} (F_{η_{V}})$ in the displayed sum is a well-defined nonnegative integer, and only the $d$ -dimensional components of $Supp F$ carry nonzero coefficients, by [L4]. The family of $d$ -dimensional components of a closed subset of a locally Noetherian scheme is locally finite, and finite when $X$ is quasi-compact, so the sum is locally finite in general and finite for finite type $X$ ; this defines $[F]_{d}$ in the sense of [L1].

2.1L3step 1.1given

Additivity. Let $0 \to F^{'} \to F \to F^{''} \to 0$ be a short exact sequence of coherent sheaves, all supported in dimension at most $d$ . For every $d$ -dimensional integral closed subscheme $V$ with generic point $η_{V}$ , localization at $η_{V}$ is exact, giving $0 \to F_{η_{V}}^{'} \to F_{η_{V}} \to F_{η_{V}}^{''} \to 0$ ; by [L3] the lengths add. Taking coefficients, $[F]_{d} = [F^{'}]_{d} + [F^{''}]_{d}$ . The common dimension bound is used here and cannot be dropped: on a smooth integral curve $C$ with a closed point $p$ the sequence $0 \to O_{C} (- p) \to O_{C} \to O_{p} \to 0$ has full-support cycles $[C], [C], [p]$ , and $[C] \neq [C] + [p]$ in $Z_{*} (C)$ .

2.2L1L2step 1.1given

Fundamental cycles. For a closed subscheme $Y \subseteq X$ , the stalk of $O_{Y}$ at the generic point $η_{Z}$ of an irreducible component $Z$ of $Y$ is a $0$ -dimensional Noetherian local ring, the local ring of $Y$ at its generic point, and its length is finite; setting the coefficient of $Z$ in $[Y]$ to that length makes $[Y]$ well defined by [L1]. If $Y$ is pure of dimension $d$ , its components of dimension at most $d$ are exactly its irreducible components, so $[Y] = [O_{Y}]_{d}$ by the definition of the latter and [L2]. If $Y$ is reduced, the local ring at the generic point of each component is a field, of length $1$ , so every coefficient of $[Y]$ is one.

3.1L3L5step 2.2algebra

Flat pullback. Let $Y$ be pure of dimension $d$ , and let $W$ be a component of $f^{- 1} Y$ of dimension $d + r$ with generic point $η_{W}$ ; write $ζ = f (η_{W})$ . Because all fibres of $f$ have pure dimension $r$ by [L5], $ζ$ is a generic point $η_{Z}$ of a component $Z$ of $Y$ , the fibre of $f$ over $ζ$ has pure dimension $r$ , and $η_{W}$ is a generic point of that fibre. The local ring of the scheme-theoretic preimage at $η_{W}$ is $O_{f^{- 1} Y, η_{W}} ≅ O_{Y, η_{Z}} \otimes_{O_{X, η_{Z}}} O_{X^{'}, η_{W}}$ : the preimage is $Y \times_{X} X^{'}$ and since $Y ↪ X$ is closed, both sides are the quotient of $O_{X^{'}, η_{W}}$ by the extended ideal of $Y$ . Put $R = O_{X, η_{Z}}$ and $M = O_{Y, η_{Z}}$ . The module $M$ has finite length by step 2.2 and therefore has a composition series whose factors are copies of $κ (η_{Z})$ , with length $m_{Z} = ℓ_{R} (M)$ . Tensoring this series with the flat $R$ -algebra $S = O_{X^{'}, η_{W}}$ preserves exactness, and each factor contributes $S / m_{Z} S$ , which is the zero-dimensional Noetherian local ring of the fibre at its generic point. Its finite length $e_{W} = ℓ_{S} (S / m_{Z} S)$ is the generic multiplicity of $W$ in $[f^{- 1} Z]_{d + r}$ , and need not be one. By additivity of length, $ℓ_{S} (O_{f^{- 1} Y, η_{W}}) = m_{Z} e_{W}$ . Thus this coefficient equals the coefficient of $W$ in $f^{*} [Y]_{d} = \sum_{Z} m_{Z} [f^{- 1} Z]_{d + r}$ , where the sum is the linear extension of the assignment $[Z] \mapsto [f^{- 1} Z]_{d + r}$ on integral cycles. Therefore $[f^{- 1} Y]_{d + r} = f^{*} [Y]_{d}$ .

4.1L1L5step 2.2given∎

Divisors. Let $Y$ be pure of dimension $d$ and let $D$ be an effective Cartier divisor whose local equation $t$ at each point of $Y$ is a nonzerodivisor on $O_{Y}$ . Then $D ∣_{Y}$ is an effective Cartier divisor with local equation $t$ , by [L5]. No irreducible component $Z$ of $Y$ is contained in $D$ : otherwise $t$ would lie in the maximal ideal of $O_{Y, η_{Z}}$ and be nilpotent there, hence a zerodivisor, contrary to the hypothesis. Consequently $Y \cap D = V (t)$ is pure of dimension $d - 1$ , and at the generic point $η$ of each of its components one has $O_{Y \cap D, η} = O_{Y, η} / t O_{Y, η}$ with $t$ a nonzerodivisor, so this local ring has finite length and its length $ℓ (O_{Y, η} / t O_{Y, η})$ is the generic coefficient of the effective Cartier divisor $D ∣_{Y}$ . This uses the quotient length, without applying the domain order function to a possibly nonreduced local ring. Hence the cycle of $D ∣_{Y}$ is the fundamental $(d - 1)$ -cycle of $Y \cap D$ as defined in step 2.2. The hypothesis cannot be weakened to $Y \subseteq̸ D$ : for $X = A^{2}$ , $Y = Spec k [x, y] / (y^{2}, x y)$ , the $x$ -axis with an embedded point at the origin, and $D = V (x)$ , one has $Y \subseteq̸ D$ but $x y = 0$ with $y \neq 0$ in $O_{Y}$ shows that $x$ is a zerodivisor, so $D ∣_{Y}$ is not an effective Cartier divisor at the embedded point.

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

Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the ample/global-generation and regular-local homological suppliers. Let $X$ be a regular quasi-projective scheme of finite type over a field. Then the natural map $K^{0} (X) ⟶ K_{0} (X), [E] ⟼ [E]$ is an isomorphism. Every coherent sheaf has a finite resolution by finite locally free sheaves, and its class corresponds to the alternating sum of such a resolution, independent of the resolution. Consequently this identification applies to every smooth quasi-projective variety used in the Riemann-Roch theorem of this page. Regularity alone is not asserted to supply a global vector-bundle resolution on an arbitrary scheme.

Facts & Assumptions

Given: the Axiom of Choice; a regular quasi-projective scheme $X$ of finite type over a field, with $n = \dim X$ ; a coherent $O_{X}$ -module $F$ .

[F1]

$X$ is Noetherian and every coherent $O_{X}$ -module is a quasi-coherent sheaf of finite type; kernels, images and cokernels of maps of coherent sheaves are coherent, and $O_{X, x}$ is a regular local ring of dimension at most $n$ at every point (Locally Noetherian and Noetherian schemes, Coherent sheaves on a locally Noetherian scheme, Coherent module sheaves, regular local residue field projective dimension dimension, embedding dimension and regular local ring).

[F2]

Because $X$ is quasi-projective over a field, the restriction $L = O (1) ∣_{X}$ of the corresponding very ample invertible sheaf is ample (Quasi-projective morphisms before Proj, Absolute ampleness by affine section opens, Relative very ampleness implies relative ampleness). For every coherent $F$ there is $ν_{0}$ such that $F \otimes L^{\otimes ν}$ is globally generated for all $ν \geq ν_{0}$ (Serre global-generation criterion for ampleness, Global generation by the evaluation map).

[F3]

A finitely generated module over a Noetherian local ring has finite projective dimension at most the dimension of the ring when the ring is regular: the global dimension of a regular local ring equals its dimension (local global dimension equals residue field projective dimension, regular local residue field projective dimension dimension). A finitely generated module over a Noetherian ring whose $n$ -th syzygy is projective has projective dimension at most $n$ at the corresponding prime (Projective dimension at most n iff the nth syzygy is projective). A finitely presented module over a local ring is free if and only if it is projective, and a coherent sheaf whose stalks are free is finite locally free (Locally free sheaves of finite rank).

[F4]

$K_{0} (X)$ and $K^{0} (X)$ are the Grothendieck groups of coherent sheaves and of finite locally free sheaves, with the evident generating classes and exact-sequence relations, and the natural comparison map is additive on classes (Grothendieck groups of coherent sheaves and of vector bundles on a scheme).

[F5]

A smooth finite type scheme over a field is regular; in particular the smooth quasi-projective varieties of the Riemann-Roch statement fall under the present hypotheses (Relative Jacobian criterion with its presentation hypothesis, Locally standard smooth iff flat with geometrically regular fibres): a smooth chart is standard smooth, hence geometrically regular, and taking the original field shows its local rings are regular.

Proof

technique · direct; produce bounded locally free resolutions using amplitude and the homological dimension bound, compare two resolutions through a common refinement, and derive additivity from a short exact sequence of resolutions

1.1F2givenalgebra

Surjections from locally free sheaves. By [F2] there is an integer $ν$ with $F \otimes L^{\otimes ν}$ globally generated. Since $X$ is quasi-compact (finite type over a field) and quasi-separated, finitely many global sections $s_{0}, \dots, s_{r}$ generate $F \otimes L^{\otimes ν}$ : the cokernel of the map $⨁_{i = 0}^{r} O_{X} \to F \otimes L^{\otimes ν}$ defined by the $s_{i}$ vanishes on a neighbourhood of each point for a suitable finite selection, and finitely many such neighbourhoods cover $X$ . Untwisting by $L^{- ν}$ gives a surjection $E_{0} : = ⨁_{i = 0}^{r} L^{- ν} ↠ F$ from a finite locally free sheaf.

2.1F1F3step 1.1algebra

Bounded locally free resolutions. Define coherent subsheaves $K_{0} = F$ and $K_{i + 1} = \ker (E_{i} \to K_{i})$ for the surjections produced by step 1.1, so that $0 \to K_{i + 1} \to E_{i} \to K_{i} \to 0$ is exact with $E_{i}$ finite locally free; all $K_{i}$ are coherent by [F1]. At a point $x$ , the local ring $R = O_{X, x}$ is regular of dimension at most $n$ by [F1], so $R$ has global dimension at most $n$ by [F3]; hence the $n$ -th syzygy of the stalk $F_{x}$ is projective over $R$ , and therefore free, because a finitely generated projective module over a local ring is free by [F3]. Since this holds at every point, $K_{n}$ is a coherent sheaf with free stalks, hence finite locally free by [F3]. Thus $F$ admits the finite locally free resolution $0 \to K_{n} \to E_{n - 1} \to \dots \to E_{0} \to F \to 0$ .

3.1F4step 1.1step 2.1algebra

Independence of the resolution. Let $E_{∙}, F_{∙}$ be bounded locally free resolutions of the same coherent sheaf $H$ . Choose $N$ at least their lengths and $n$ , padding by zero terms. Construct a resolution $G_{∙}$ with degreewise surjective maps to both. Start with $K_{0} (G) = K_{0} (E) = K_{0} (F) = H$ . If $K_{i} (G)$ surjects onto $K_{i} (E), K_{i} (F)$ , form the coherent sheaf $P_{i} = (E_{i} \times_{K_{i} (E)} K_{i} (G)) \times_{K_{i} (G)} (F_{i} \times_{K_{i} (F)} K_{i} (G))$ . Its projections onto $E_{i}, F_{i}, K_{i} (G)$ are surjective by local lifting through the given surjections. Cover $P_{i}$ by a finite locally free $G_{i}$ using step 1.1. Taking augmentation kernels gives surjections $K_{i + 1} (G) \to K_{i + 1} (E), K_{i + 1} (F)$ by the kernel calculation in a diagram of short exact sequences. At degree $N$ use $G_{N} = K_{N} (G)$ , locally free by the dimension bound in step 2.1, mapping onto the terminal syzygies $E_{N}, F_{N}$ . The kernel complexes of $G_{∙} \to E_{∙}, F_{∙}$ are bounded acyclic complexes of locally free sheaves: degreewise surjections between locally free sheaves split locally. Such a complex has zero alternating class, because starting at its lowest degree its successive cycle sheaves are locally free and the resulting short exact sequences telescope. The alternating classes of $E_{∙}, F_{∙}, G_{∙}$ therefore agree.

4.1F1F4step 1.1step 3.1algebra

Additivity. Given $0 \to H^{'} \to H \to H^{''} \to 0$ , construct compatible resolutions term by term. Put $K_{0}^{'} = H^{'}$ , $K_{0} = H$ , $K_{0}^{''} = H^{''}$ . Suppose $0 \to K_{i}^{'} \to K_{i} \to K_{i}^{''} \to 0$ is exact. Choose locally free covers $C_{i} \to K_{i}^{''}$ and $A_{i} \to K_{i} \times_{K_{i}^{''}} C_{i}$ by step 1.1. The composite $A_{i} \to C_{i}$ surjects, so its kernel $B_{i}$ is locally free; the induced $B_{i} \to K_{i}^{'}$ also surjects, by local lifting. Taking the kernels of the three augmentations yields $0 \to K_{i + 1}^{'} \to K_{i + 1} \to K_{i + 1}^{''} \to 0$ . After $n$ stages all three syzygies are locally free by the dimension bound; take them as terminal terms. This produces a short exact sequence of bounded locally free resolutions of $H^{'}, H, H^{''}$ . Alternating classes add term by term, and independence in step 3.1 gives additivity for every choice of resolutions.

5.1F4F5step 3.1step 4.1∎

The comparison isomorphism. Both maps are well defined and additive: the natural map $K^{0} (X) \to K_{0} (X)$ sends the class of a finite locally free sheaf to its class in $K_{0} (X)$ and respects the exact-sequence relations of [F4], while the assignment sending the class of a coherent sheaf $H$ to the alternating class of any bounded locally free resolution of $H$ is well defined by step 3.1 and additive by step 4.1, hence descends to a homomorphism $K_{0} (X) \to K^{0} (X)$ . The composite $K^{0} (X) \to K_{0} (X) \to K^{0} (X)$ is the identity because a finite locally free sheaf is its own length-zero resolution, and the composite $K_{0} (X) \to K^{0} (X) \to K_{0} (X)$ is the identity because the alternating class of a resolution of $H$ maps to $[H]$ in $K_{0} (X)$ by the telescoping exact-sequence relations. Therefore the natural map is an isomorphism, and in particular it applies to every smooth quasi-projective variety over a field, which is regular and quasi-projective by [F5].

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

Pushforward of coherent sheaves in algebraic K-theory

Definition

Assume the Axiom of Choice and the Axiom of Dependent Choice, inherited from the coherence and finiteness suppliers below (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 a field and let $f : X \to Y$ be a proper morphism of schemes of finite type over $k$ (Proper morphisms, Locally finite type and finite type morphisms); then $Y$ is locally Noetherian and for every coherent $O_{X}$ -module $F$ all higher direct images $R^{q} f_{*} F$ are coherent $O_{Y}$ -modules (Coherent higher direct images under proper morphisms, Higher direct image of a sheaf) and vanish for all sufficiently large $q$ (Dimension bound for quasi-coherent cohomology on a Noetherian scheme on the separated Noetherian preimage of each affine open of $Y$ , giving the single global bound $q > \dim X$ ). Set $f_{!} [F] : = \sum_{q \geq 0} (- 1)^{q} [R^{q} f_{*} F] \in K_{0} (Y)$ (Grothendieck groups of coherent sheaves and of vector bundles on a scheme). Then:

$f_{!}$ is a well-defined homomorphism $K_{0} (X) \to K_{0} (Y)$ of abelian groups: a short exact sequence $0 \to F^{'} \to F \to F^{''} \to 0$ on $X$ induces the long exact sequence of higher direct images $0 \to f_{*} F^{'} \to f_{*} F \to f_{*} F^{''} \to R^{1} f_{*} F^{'} \to \dots$ (Higher direct image of a sheaf, The derived long exact sequence), whose alternating sum is zero, giving $f_{!} [F] = f_{!} [F^{'}] + f_{!} [F^{''}]$ .
(Functoriality) ${id}_{!} = id$ and $(g \circ f)_{!} = g_{!} \circ f_{!}$ for composable proper morphisms: the bounded Leray spectral sequence for the composition preserves its total alternating class on every page; no degeneration or splitting is asserted (Leray spectral sequence for sheaf cohomology).
(Compatibility with the module structure) $f_{!} (f^{*} [E] \cdot [F]) = [E] \cdot f_{!} [F]$ for $E$ locally free on $Y$ and $F$ coherent on $X$ , i.e. the projection formula; this is proved in lem-projection-formula-in-algebraic-k-theory. For $Y = Spec k$ and $X$ projective, $f_{!} [F] = χ (X, F)$ is the Euler characteristic (Euler characteristic of a coherent sheaf, Euler characteristic is additive in short exact sequences).

Well-definedness. The sum in the definition is finite because $R^{q} f_{*} F = 0$ for $q > \dim X$ : for an affine open $U \subseteq Y$ the preimage $f^{- 1} U$ is separated, Noetherian and of dimension at most $\dim X$ , and Dimension bound for quasi-coherent cohomology on a Noetherian scheme bounds its quasi-coherent cohomology uniformly, the higher direct image being the sheafification of these local cohomology groups (Cech cohomology computes quasi-coherent cohomology on a separated scheme). Additivity in short exact sequences is the alternating-sum cancellation in the bounded long exact sequence of higher direct images, together with the coherence of every term (Coherent higher direct images under proper morphisms, Grothendieck groups of coherent sheaves and of vector bundles on a scheme). For the composition spectral sequence apply Grothendieck spectral sequence to $f_{*}$ and $g_{*}$ on module sheaves. Module-injective sheaves are flasque (Injective modules are flasque and Ext from the structure sheaf is cohomology), their direct images are flasque, and flasque sheaves are acyclic on every open (Flasque abelian sheaves are Γ-acyclic). As in the module/abelian comparison in Leray spectral sequence for sheaf cohomology, this makes them $g_{*}$ -acyclic and supplies the required acyclicity hypothesis. Functoriality is then page-invariance of the total alternating class: for composable proper $f : X \to Y$ , $g : Y \to Z$ the $E_{2}$ page $R^{p} g_{*} (R^{q} f_{*} F)$ is bounded in both directions by the uniform dimension bounds for $f$ , $g$ and $g f$ , each differential changes total degree by one so that the two kernel/image short exact sequences cancel every image class, and the finite abutment filtration identifies the invariant with the alternating class of $R^{∙} (g \circ f)_{*} F$ (Leray spectral sequence for sheaf cohomology); no degeneration is used. The projection formula is proved separately in lem-projection-formula-in-algebraic-k-theory, and for $Y = Spec k$ the definition reduces to the Euler characteristic.

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

Proper pushforward of cycles and the norm formula

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Let $k$ be a field and let $f : X \to Y$ be a proper morphism of schemes locally of finite type over $k$ (Proper morphisms, Locally finite type and finite type morphisms). For an integral closed subscheme $V \subseteq X$ with generic point $η$ , let $W = \overline{f (V)} \subseteq Y$ with the reduced structure, an integral closed subscheme (Scheme-theoretic image, Proper morphisms are closed). Define $f_{*} [V] : = {\begin{cases} [k (V) : k (W)] \cdot [W], & \dim W = \dim V, \\ 0, & \dim W < \dim V, \end{cases}$ where $k (V)$ , $k (W)$ are the function fields (Sheaf total quotient rings) and $[k (V) : k (W)]$ is the extension degree (The degree $[K : F] = \dim_{F} K$ of a finite field extension), finite because a dominant morphism $V \to W$ of integral finite-type $k$ -schemes with $\dim V = \dim W$ has algebraic, hence finite, function field extension. Extending $Z$ -linearly gives $f_{*} : Z_{d} (X) \to Z_{d} (Y)$ for all $d$ . Then:

$f_{*}$ is a homomorphism of graded groups and $f_{*} ({Rat}_{d} (X)) \subseteq {Rat}_{d} (Y)$ , so it descends to $f_{*} : A_{d} (X) \to A_{d} (Y)$ (Rational equivalence and the Chow group of cycles).
(Functoriality) ${id}_{*} = id$ and $(g \circ f)_{*} = g_{*} \circ f_{*}$ for proper $f, g$ ; the degree is multiplicative in towers of finite function-field extensions.
(Base change) For a cartesian square with $g$ flat of pure relative dimension $n$ and proper $f$ , the compatibility of lem-pushforward-pullback-compatibility-chow holds.
(Normalization) If $f$ is finite flat of constant degree $n$ between integral schemes of the same dimension, then $f_{*} [X] = n [Y]$ .

Point (1) is the nontrivial assertion: for $W \subseteq X$ integral of dimension $d + 1$ and $r \in k (W)^{*}$ one has $f_{*} ({div}_{W} (r)) = {div}_{f (W)} (N_{k (W) / k (f (W))} (r))$ when $\dim f (W) = \dim W$ , and $f_{*} ({div}_{W} (r)) = 0$ when $\dim f (W) < \dim W$ ; here $N$ is the field norm.

Facts & Assumptions

Given: the Axiom of Choice; a proper morphism $f : X \to Y$ of schemes locally of finite type over $k$ ; an integral closed subscheme $V \subseteq X$ with function field $L = k (V)$ and image $W$ with function field $K = k (W)$ .

[F1]

$V$ is integral and locally of finite type over $k$ ; each nonempty affine chart is of finite type and has the same function field, $W$ is integral of dimension at most $\dim V$ , and $L / K$ is finitely generated; if $\dim W = \dim V$ then $L / K$ is algebraic, hence finite of degree $[L : K]$ , by the dimension-transcendence-degree theorem (Integral schemes, Proper morphisms are closed, Scheme-theoretic image, Affine-domain dimension equals transcendence degree, The degree $[K : F] = \dim_{F} K$ of a finite field extension). A dense open subscheme of $V$ has the same function field, and the relative algebraic-constants lemma supplies the finiteness statements for dominant finite-type morphisms of integral schemes (A finite-type field has finite relative algebraic constants).

[F2]

Cycles and rational equivalence: $Z_{d}$ and ${Rat}_{d}$ are as in Algebraic cycles and the cycle group of a scheme of finite type over a field and Rational equivalence and the Chow group of cycles, with divisor cycles ${div}_{W} (r) = \sum_{Z} {ord}_{O_{W, Z}} (r) [Z]$ computed by the order function of The order function of a one-dimensional Noetherian local domain; the order function is multiplicative, additive on products and normalized so that $ord$ is the valuation on a discrete valuation ring.

[F3]

Length is additive in short exact sequences and a finitely generated module over a one-dimensional Noetherian local domain has finite length after quotient by a nonzerodivisor (Module length is additive in short exact sequences, The order function of a one-dimensional Noetherian local domain).

[F4]

A proper quasi-finite morphism is finite, and the quasi-finite locus of a finite type morphism is open (A proper quasi-finite morphism is finite, The quasi-finite locus of a finite-type algebra is open).

Proof

technique · direct; define the pushforward of cycles, prove the norm formula by a finite lattice-index computation, and handle the dimension-drop cases by the degree of principal divisors on proper curves

1.1F1F2given

The cycle pushforward. The closure $W$ of $f (V)$ is closed by properness and irreducible, and we give it the reduced structure, so $W$ is integral with function field $K$ ; $\dim W \leq \dim V$ and $[L : K]$ is finite when $\dim W = \dim V$ by [F1]. The displayed formula therefore defines a graded homomorphism $f_{*} : Z_{d} (X) \to Z_{d} (Y)$ , since integral closed subschemes of a fixed dimension form a basis of $Z_{d}$ and the coefficient is an integer. For a dense open $U \subseteq V$ one has $\overline{f (U)} = W$ and $k (U) = k (V)$ , so the definition is insensitive to replacing $V$ by a dense open.

1.2F2F3algebra

The finite norm-order formula. Let $(A, m)$ be a one-dimensional Noetherian local domain with fraction field $K$ , and let $B$ be a finite domain over $A$ with fraction field $L$ , finite over $K$ . The ring $B$ is semilocal; for $r \in L^{*}$ , ${ord}_{A} (N_{L / K} (r)) = \sum_{q \in Max (B)} [κ (q) : κ (A)] {ord}_{B_{q}} (r) .$ To prove this, call a finite torsion-free $A$ -submodule of $L$ spanning $L$ a full lattice. Any two full lattices are commensurable, so for such lattices set $δ (M, N) = ℓ_{A} (M / (M \cap N)) - ℓ_{A} (N / (M \cap N)) .$ Length additivity makes $δ$ additive in chains of lattices; a $K$ -linear isomorphism preserves it. Consequently $g \mapsto δ (M, g M)$ is a homomorphism ${GL}_{n} (K) \to Z$ , independent of the chosen full lattice $M$ , where $n = [L : K]$ . For $M = A^{n}$ , a diagonal matrix has index the sum of the orders of its diagonal entries. For an elementary transvection $E_{i j} (u)$ with $u \in K$ , put $I = {a \in A : u a \in A}$ . The intersection of $A^{n}$ and $E_{i j} (u) A^{n}$ has the same coordinates as $A^{n}$ except for the $j$ th coordinate, which is $I$ ; both quotients by this intersection are isomorphic to $A / I$ , so the index is zero. Gaussian elimination therefore gives $δ (M, g M) = {ord}_{A} (\det g)$ for every $g \in {GL}_{n} (K)$ . Apply this to multiplication by $r$ on the lattice $B$ : its determinant is $N_{L / K} (r)$ . Write $r = a / b$ with nonzero $a, b \in B$ . Translation by $b$ and additivity of $δ$ give ${ord}_{A} (N_{L / K} (r)) = ℓ_{A} (B / a B) - ℓ_{A} (B / b B) .$ For any nonzero $a \in B$ , the quotient $B / a B$ is a zero-dimensional Noetherian ring, hence has finite length; as a $B$ -module it has a composition series whose simple factors are the residue fields at maximal ideals of $B$ . Thus $ℓ_{A} (B / a B) = \sum_{q} [κ (q) : κ (A)] ℓ_{B_{q}} (B_{q} / a B_{q}) .$ Subtracting the analogous equality for $b$ proves the formula by the definition of the order function.

2.1F4F1step 1.2algebra

Equal-dimensional images. Suppose $\dim W = \dim V = d + 1$ . Let $T \subseteq W$ be an integral closed subscheme of dimension $d$ , with generic point $ζ$ . The fibre of $V \to W$ over $ζ$ is zero-dimensional: a positive-dimensional component would have closure of dimension at least $d + 1$ in the integral scheme $V$ , hence would be all of $V$ , contradicting dominance. Thus $V \to W$ is quasi-finite at every point over $ζ$ . Its quasi-finite locus is open, and properness lets us shrink around $ζ$ so that the restriction is proper quasi-finite, hence finite by [F4]. The resulting finite algebra over $A = O_{W, ζ}$ is a domain finite over $A$ , and its maximal ideals correspond to the codimension-one subschemes of $V$ mapping onto $T$ . The formula in step 1.2 says that the coefficient of $[T]$ in ${div}_{W} (N_{k (V) / k (W)} (r))$ is $\sum_{Z \mapsto T} [k (Z) : k (T)] {ord}_{O_{V, Z}} (r),$ which is exactly the coefficient of $[T]$ in $f_{*} {div}_{V} (r)$ . Codimension-one subschemes of $V$ whose image has dimension less than $d$ push forward to zero and do not contribute to the divisor on $W$ . Equality at every such $T$ proves $f_{*} {div}_{V} (r) = {div}_{W} (N_{k (V) / k (W)} (r)) .$ .

2.2F1step 1.1algebra

Functoriality. For the identity morphism the formula is $[k (V) : k (V)] [V] = [V]$ . For proper composable $f : X \to Y$ , $g : Y \to Z$ and an integral $V \subseteq X$ with closure $W = \overline{f (V)}$ and closure $T = \overline{g (W)}$ , the function field degrees multiply in the tower $k (T) \subseteq k (W) \subseteq k (V)$ when all three dimensions agree, both sides give $[k (V) : k (T)] [T]$ , and if either dimension drops then both composites give zero; a proper morphism carries a locally finite family of integral closed subschemes to a locally finite family, so the identity extends to locally finite cycles.

3.1F1F2step 2.1algebra

Dimension drop. If $\dim W \leq \dim V - 2$ , every codimension-one subscheme $Z \subset V$ has dimension $\dim V - 1$ , while $\dim f (Z) \leq \dim W \leq \dim V - 2 < \dim Z$ ; hence its pushforward is zero. Suppose instead that $\dim V = d + 1$ and $\dim W = d$ . The generic fibre $C = V \times_{W} Spec k (W)$ is a proper integral curve over $K = k (W)$ . Codimension-one subschemes of $V$ that dominate $W$ correspond to closed points $x$ of $C$ , and their contribution to the coefficient of $[W]$ in $f_{*} {div}_{V} (r)$ is $\sum_{x \in C closed} [k (x) : K] {ord}_{O_{C, x}} (r) .$ This degree is zero. Indeed, if $r$ is constant its divisor is zero. Otherwise let $Γ$ be the closure of the graph of the rational map $r : C ⇢ P_{K}^{1}$ . The projections $p : Γ \to C$ and $q : Γ \to P_{K}^{1}$ are proper; $p$ is birational, while $q$ is nonconstant, hence quasi-finite and finite. Since the local rings of $P_{K}^{1}$ are fields or discrete valuation rings and $Γ$ is integral, the finite morphism $q$ is flat of some degree $e$ . The equal-dimensional formula of step 2.1 gives $p_{*} {div}_{Γ} (r) = {div}_{C} (r)$ , and ${div}_{Γ} (r) = q^{*} [0] - q^{*} [\infty]$ . Both fibres have degree $e$ over $K$ , so the displayed sum is $e - e = 0$ . This proves $f_{*} {div}_{V} (r) = 0$ in the remaining case as well.

4.1F1step 1.1givenalgebra∎

Base change and finite flat normalization. Consider a cartesian square with $f : X \to Y$ proper and $g : Y^{'} \to Y$ flat of pure relative dimension $n$ , and write $f^{'} : X^{'} = X \times_{Y} Y^{'} \to Y^{'}$ and $g^{'} : X^{'} \to X$ . For an integral $d$ -dimensional $V \subseteq X$ , put $W = \overline{f (V)}$ and $e = \dim V - \dim W$ . If $e > 0$ , both $g^{*} f_{*} [V]$ and $f_{*}^{'} g^{' *} [V]$ are zero: every component of the flat pullback of $V$ has dimension $d + n$ , while its image has dimension at most $\dim W + n < d + n$ . If $e = 0$ , set $m = [k (V) : k (W)]$ . For each component $W_{i}$ of $g^{- 1} (W)$ , let $C_{i}$ be the Artinian local ring of $W \times_{Y} Y^{'}$ at its generic point. Flat pullback gives coefficient $ℓ (C_{i})$ on $[W_{i}]$ . The generic algebra of $V \times_{Y} Y^{'}$ over $C_{i}$ is $C_{i} \otimes_{k (W)} k (V)$ , a free $C_{i}$ -module of rank $m$ . Decomposing it into its Artinian local factors shows that the sum of the generic lengths of the components above $W_{i}$ , weighted by their function-field degrees over $k (W_{i})$ , is $m ℓ (C_{i})$ . These are precisely the coefficients of $[W_{i}]$ in $f_{*}^{'} g^{' *} [V]$ and $g^{*} f_{*} [V]$ , respectively. Hence the base-change identity holds on cycles and therefore on Chow groups. Finally, if $f$ is finite flat of constant degree $n$ between integral schemes of the same dimension, then $f$ is dominant and $[k (X) : k (Y)] = n$ , so $f_{*} [X] = n [Y]$ by step 1.1.

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

Projection formula for higher direct images and K-theory pushforward

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $f : X \to Y$ be a proper morphism of schemes of finite type over a field $k$ (Proper morphisms, Locally finite type and finite type morphisms). Let $E$ be a finite locally free $O_{Y}$ -module (Locally free sheaves of finite rank) and let $F$ be a coherent $O_{X}$ -module (Coherent module sheaves). Then for every $q \geq 0$ there is a canonical isomorphism of coherent $O_{Y}$ -modules $R^{q} f_{*} (f^{*} E \otimes_{O_{X}} F) ≅ E \otimes_{O_{Y}} R^{q} f_{*} F,$ and consequently, in $K_{0} (Y)$ (Grothendieck groups of coherent sheaves and of vector bundles on a scheme), $f_{!} (f^{*} [E] \cdot [F]) = [E] \cdot f_{!} [F],$ where $f_{!}$ is the pushforward of Pushforward of coherent sheaves in algebraic K-theory and the products are the $K$ -theory module products.

Facts & Assumptions

Given: the Axiom of Choice; a proper morphism $f : X \to Y$ of finite type $k$ -schemes; a finite locally free $O_{Y}$ -module $E$ ; a coherent $O_{X}$ -module $F$ .

[F1]

The higher direct images $R^{q} f_{*} F$ are coherent and vanish for $q > \dim X$ ; hence the alternating sum $f_{!} [F]$ is a well-defined element of $K_{0} (Y)$ (Pushforward of coherent sheaves in algebraic K-theory, Grothendieck groups of coherent sheaves and of vector bundles on a scheme).

[F2]

Pullback of quasi-coherent sheaves is quasi-coherent, tensor products of quasi-coherent sheaves are quasi-coherent, and pullback of a finite locally free sheaf is finite locally free (Tensor product preserves quasi-coherence, Locally free sheaves of finite rank).

[F3]

For a finitely presented quasi-coherent sheaf $E$ and a quasi-coherent sheaf $G$ on $Y$ , the internal Hom is quasi-coherent and its affine description is the sheaf associated to the module of linear maps; these descriptions agree on overlaps (Internal Hom from a finitely presented sheaf is quasi-coherent). The higher-cohomology comparison below is constructed directly from the tensor maps attached to sections of $E$ .

Proof

technique · direct; reduce to the structure sheaf on a trivializing affine cover, then telescope over the finite algebraic local trivializations

1.1F2F3givenalgebra

Canonical map and the locally free case. For an open $U \subseteq Y$ and a section $e$ of $E ∣_{U}$ , the morphism $F ∣_{f^{- 1} U} \to (f^{*} E \otimes F) ∣_{f^{- 1} U}$ sending $s$ to $f^{*} e \otimes s$ induces a map on every higher direct image. This construction is additive in $e$ , linear over $O_{Y} (U)$ , and compatible with restriction. Tensoring and sheafifying therefore defines the canonical comparison $E \otimes R^{q} f_{*} F \to R^{q} f_{*} (f^{*} E \otimes F)$ . Suppose first that $E = O_{Y}^{d}$ is free. Then $f^{*} E = O_{X}^{d}$ , tensoring with it commutes with the direct image and with cohomology, and the map is the identity componentwise; so the comparison map is an isomorphism in this case.

2.1F2step 1.1algebra

Local trivialization and additivity. Since $E$ is finite locally free, $Y$ is covered by affine opens $U$ on which $E$ is free, and on each such $U$ the restriction of the comparison map is an isomorphism by step 1.1, using that the restrictions of $f$ , $E$ and $F$ compute the restricted higher direct images. Two maps of quasi-coherent sheaves that agree on an open cover agree, so the comparison map is an isomorphism for every finite locally free $E$ : the argument is local on $Y$ and the local triviality makes the free case available, so no further additivity over direct summands is needed.

3.1F1F2step 2.1algebra∎

The K-theory identity. Tensoring a coherent sheaf with a finite locally free sheaf is exact, so on $K_{0}$ the class $[E]$ acts by $[G] \mapsto [E \otimes G]$ , and $f^{*} [E]$ is a finite locally free class by [F2]. Summing the isomorphisms of step 2.1 with alternating signs gives $f_{!} (f^{*} [E] \cdot [F]) = \sum_{q} (- 1)^{q} [E \otimes R^{q} f_{*} F]$ in $K_{0} (Y)$ by [F1], and $\sum_{q} (- 1)^{q} [E \otimes R^{q} f_{*} F] = [E] \cdot \sum_{q} (- 1)^{q} [R^{q} f_{*} F] = [E] \cdot f_{!} [F]$ because the action of $[E]$ is additive. That is the projection formula in $K_{0} (Y)$ .

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

K-theory of projective space and of projective bundles

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field.

In $K^{0} (P_{k}^{n}) = K_{0} (P_{k}^{n})$ (Grothendieck groups of coherent sheaves and of vector bundles on a scheme, Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes) the classes $[O (i)]$ , $0 \leq i \leq n$ , form a $Z$ -basis; equivalently $K_{0} (P^{n}) = Z^{n + 1}$ generated by the twisting sheaves (Twisting sheaf on Proj, Invertible twists for degree-one generated rings).
For every locally Noetherian $k$ -scheme $Y$ and every $n \geq 0$ , the exterior-product map $K_{0} (Y) \otimes_{Z} K_{0} (P_{k}^{n}) ⟶ K_{0} (P_{k}^{n} \times_{k} Y)$ is surjective; here $K_{0} (P^{n} \times_{k} Y)$ is the Grothendieck group of coherent sheaves (Grothendieck groups of coherent sheaves and of vector bundles on a scheme) and the product is $[F] \times [G] \mapsto [p_{Y}^{*} F \otimes p_{P}^{*} G]$ (Tensor product preserves quasi-coherence, Scheme pullback preserves quasi-coherence).
More generally, if $E$ is a finite locally free $O_{Y}$ -module of rank $r + 1$ , the projective bundle $P (E)$ has $K_{0} (P (E))$ equal to the image of $⨁_{a = 0}^{r} K_{0} (Y)$ under $(α_{a}) \mapsto \sum_{a} q^{*} α_{a} \otimes [O (a)]$ ; this is generation by twists, without treating coherent $K_{0} (Y)$ as a tensor-product ring on a singular base, via the relative-diagonal Koszul computation; no global cell decomposition of a nontrivial bundle is assumed (Projective bundle in the quotient convention).

Facts & Assumptions

Given: the Axiom of Choice; a field $k$ ; a locally Noetherian $k$ -scheme $Y$ ; a finite locally free $O_{Y}$ -module $E$ of rank $r + 1 \geq 1$ ; the projective bundle $q : P = P (E) \to Y$ of one-dimensional quotients of $E$ with twisting sheaf $O (1)$ and universal exact sequence $0 \to S \to q^{*} E \to O (1) \to 0$ , where $S$ is the tautological subbundle of rank $r$ .

[F1]

$q$ is proper, flat and smooth of relative dimension $r$ , and its fibres are projective spaces of dimension $r$ ; the twisting sheaves $O (a)$ are invertible, and on $P_{k}^{n}$ the standard affine Čech cover computes their cohomology (Projective bundle in the quotient convention, Twisting sheaf on Proj, Invertible twists for degree-one generated rings, Cech cohomology computes quasi-coherent cohomology on a separated scheme).

[F2]

On $P_{A}^{n}$ with $A$ a commutative ring and $d \in Z$ , $H^{q} = 0$ unless $q = 0$ or $q = n$ ; $H^{0} (O (d)) = A [x_{0}, \dots, x_{n}]_{d}$ for $d \geq 0$ and $0$ for $d < 0$ when $n > 0$ ; so for the Euler characteristic over a field $χ (O (d)) = (\binom{n + d}{n})$ for $d \geq 0$ and $χ (O (d)) = 0$ for $- n \leq d \leq - 1$ (Cohomology of O(d) on projective space).

[F3]

Pullback of quasi-coherent sheaves is quasi-coherent and tensor products of quasi-coherent sheaves are quasi-coherent; exterior powers and duals of finite locally free sheaves are finite locally free, and the algebraic exterior powers are formed locally as the alternating quotient of the tensor algebra and glued by the exterior powers of the transition matrices. A locally split short exact sequence has the usual exterior-power filtration: with a line quotient its two graded pieces are $⋀^{j} S$ and $⋀^{j - 1} S \otimes L$ , as follows directly in a local basis (Scheme pullback preserves quasi-coherence, Tensor product preserves quasi-coherence, Exterior Algebra Of A Finite Free Module).

[F4]

Projection formula: for a morphism $f : X \to Y$ , a quasi-coherent $F$ on $X$ and an invertible $L$ on $Y$ , $R^{q} f_{*} (F) \otimes L ≅ R^{q} f_{*} (F \otimes f^{*} L)$ (Projection formula for invertible twists). On a locally Noetherian scheme the higher direct images of a coherent sheaf under a proper morphism are coherent, so classes of pushforwards of coherent sheaves lie in $K_{0}$ (Coherent higher direct images under proper morphisms, Grothendieck groups of coherent sheaves and of vector bundles on a scheme).

[F5]

On a regular quasi-projective scheme of finite type over a field, $K^{0} = K_{0}$ via finite locally free resolutions (Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes).

Proof

technique · direct; resolve the diagonal of $P\times_YP$ by the Koszul complex of its defining regular section, push forward, and read generation and independence off the resulting class identity

1.1F1F3givenalgebra

The diagonal section. On $P \times_{Y} P$ use the quotient convention of [F1], with $S = \ker (q^{*} E \to O (1))$ of rank $r$ . The composite $S_{2} ↪ q^{*} E ↠ O (1)_{1}$ is a section of $S_{2}^{\lor} \otimes O (1)_{1}$ ; its zero scheme is the diagonal, because the first quotient then factors through the second quotient and the resulting surjection of invertible sheaves is an isomorphism. On a common standard chart where the same coordinate of both quotients is nonzero, the section has regular equations $x_{i} - y_{i}$ , $1 \leq i \leq r$ . These are a regular sequence over every base ring, successively eliminating one coordinate. Such charts cover the diagonal, and off the diagonal one component of the section is a unit so the Koszul complex is contractible. The Koszul complex therefore resolves $O_{Δ}$ globally, with terms $O (- j)_{1} \otimes ⋀^{j} S_{2}$ .

2.1F1F3F4step 1.1algebra

Pushing the diagonal forward. Tensor the diagonal Koszul resolution with ${pr}_{1}^{*} F$ for a coherent sheaf $F$ on $P$ . It remains exact: near the diagonal the equations are coordinate differences in the second factor, a regular sequence on the polynomial extension of every module from the first factor; away from the diagonal it is contractible. Its diagonal term is $Δ_{*} F$ . Apply the alternating coherent pushforward along the proper projection ${pr}_{2}$ . On an affine open of $Y$ trivializing $E$ , the standard projective Čech complex is bounded of length $r$ and computes all higher direct images; tensoring that complex by a flat algebra commutes with its cohomology. This proves flat base change for the flat map $q$ used here. The projection formula for a locally free factor follows on trivializing opens by finite-direct-sum compatibility of cohomology, as in [F4]. Hence $[F] = \sum_{j = 0}^{r} (- 1)^{j} q^{*} q_{!} (F (- j)) \otimes [⋀^{j} S]$ . The higher direct images entering $q_{!}$ are coherent for the proper morphism $q$ and vanish above $r$ by this Čech calculation; the identity is valid on a locally Noetherian base without a ring structure on coherent $K_{0} (Y)$ .

3.1F3F4step 2.1algebra

Generation by twists. For $0 \to S \to q^{*} E \to O (1) \to 0$ , the exterior-power filtration gives $[⋀^{j} q^{*} E] = [⋀^{j} S] + [⋀^{j - 1} S] [O (1)]$ . Inducting on $j$ yields $[⋀^{j} S] = \sum_{a = 0}^{j} (- 1)^{a} [q^{*} ⋀^{j - a} E] [O (a)]$ . Substitute this into step 2.1 and absorb the pulled-back exterior powers into the coefficient in $K_{0} (Y)$ via its vector-bundle module action. Every coherent class on $P$ is therefore a sum of $q^{*} β_{a} \otimes [O (a)]$ , $0 \leq a \leq r$ , proving assertion 3.

4.1F3step 3.1algebra

The trivial bundle and the exterior product. Take $E = O_{Y}^{n + 1}$ , so that $P = P_{k}^{n} \times_{k} Y$ , the map $q$ is the second projection and $q^{*} K_{0} (Y)$ is the image of the exterior product with $[O_{P^{n}}]$ . By step 3.1 the group $K_{0} (P^{n} \times_{k} Y)$ is generated by the classes $q^{*} α \otimes [O (a)]$ with $α \in K_{0} (Y)$ and $0 \leq a \leq n$ , and each such class is the exterior product of a class on $Y$ with $[O (a)]$ on $P^{n}$ . Hence the exterior-product map $K_{0} (Y) \otimes K_{0} (P^{n}) \to K_{0} (P^{n} \times_{k} Y)$ is surjective, which is assertion 2.

5.1F2F5step 3.1algebra∎

The case $Y = Spec k$ and independence. Let $Y = Spec k$ and $E = k^{n + 1}$ , so $P = P_{k}^{n}$ . Generation by $[O (0)], \dots, [O (n)]$ is step 3.1, and $K^{0} (P) = K_{0} (P)$ by [F5] because $P_{k}^{n}$ is smooth, projective and hence regular and quasi-projective. For independence suppose $\sum_{a = 0}^{n} c_{a} [O (a)] = 0$ in $K_{0} (P)$ . Pairing with the Euler characteristic $χ (P, - \otimes O (- j))$ , for $0 \leq j \leq n$ , is additive on $K_{0}$ , and by [F2] it sends $[O (a)]$ to $χ (O (a - j)) = (\binom{n + a - j}{n})$ when $a \geq j$ and to $0$ when $a < j$ , since then $- n \leq a - j \leq - 1$ . The resulting matrix is upper triangular with diagonal entries $χ (O (0)) = 1$ , hence invertible over $Z$ , so all $c_{a} = 0$ ; therefore $[O (0)], \dots, [O (n)]$ is a basis and $K_{0} (P_{k}^{n}) ≅ Z^{n + 1}$ , which is assertion 1.

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

Flat pullback of cycles and of rational equivalence

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Let $k$ be a field and let $f : X \to Y$ be a flat morphism of schemes locally of finite type over $k$ (Flat morphism of schemes) all of whose nonempty fibres have pure dimension $n$ (Scheme-theoretic fibre, Chain dimension and the empty-space convention); for instance $f$ smooth of relative dimension $n$ (Smooth morphism of schemes, Relative dimension of a smooth morphism at a point, Fibres of a smooth morphism are smooth) or an open immersion with $n = 0$ . For an integral closed subscheme $V \subseteq Y$ let $f^{- 1} (V) = V \times_{Y} X$ be the scheme-theoretic preimage and let $[f^{- 1} (V)]$ be its cycle (Cycles of coherent sheaves and of closed subschemes, with flat pullback). Extending linearly gives $f^{*} : Z_{d} (Y) \to Z_{d + n} (X)$ . Then:

If $f^{- 1} (V)$ is nonempty, it is equidimensional of dimension $\dim V + n$ ; if it is empty, $f^{*} [V] = 0$ . In both cases $f^{*} [V]$ is a pure $(d + n)$ -cycle; this is the only place the pure-fibre-dimension hypothesis is used.
$f^{*} ({Rat}_{d} (Y)) \subseteq {Rat}_{d + n} (X)$ : for an integral closed subscheme $W \subseteq Y$ of dimension $d + 1$ and $r \in k (W)^{*}$ , if $W_{j}$ are the reduced irreducible components of $f^{- 1} (W)$ with generic lengths $n_{j}$ , then $f^{*} ({div}_{W} (r)) = \sum_{j} n_{j} {div}_{W_{j}} (r ∣_{W_{j}})$ , pushed to $X$ , and the right-hand side lies in ${Rat}_{d + n} (X)$ .
Consequently $f^{*}$ descends to a graded homomorphism $f^{*} : A_{d} (Y) \to A_{d + n} (X)$ (Rational equivalence and the Chow group of cycles).
(Functoriality) ${id}^{*} = id$ and $(g \circ f)^{*} = f^{*} \circ g^{*}$ for composable flat morphisms of the stated kind; if $f$ is an open immersion then $f^{*}$ is the restriction of cycles.
(Localization) If $j : U ↪ X$ is an open immersion and $i : Z = X ∖ U ↪ X$ the complementary reduced closed subscheme, the sequence $A_{*} (Z) \overset{i_{*}}{\to} A_{*} (X) \overset{j^{*}}{\to} A_{*} (U) \to 0$ is exact (using Proper pushforward of cycles and the norm formula for $i_{*}$ ).

Facts & Assumptions

Given: the Axiom of Choice; a flat morphism $f : X \to Y$ of schemes locally of finite type over $k$ whose nonempty fibres have pure dimension $n$ ; an integral closed subscheme $W \subseteq Y$ with function field $k (W)$ and a rational function $r \in k (W)^{*}$ .

[L1]

Under flat ring maps, minimal primes contract to minimal primes by going down. A finite-type domain over a field has dimension equal to the transcendence degree of its fraction field, and transcendence degree is additive in finite field towers. Thus if a component of a flat preimage dominates an integral base $V$ , its generic-fibre component of dimension $n$ gives total dimension $\dim V + n$ . In the smooth example, relative dimension means that every geometric fibre has the indicated local dimension and smoothness is preserved on fibres (Flat morphism of schemes, Scheme-theoretic fibre, Chain dimension and the empty-space convention, Every flat ring map satisfies going down, Affine-domain dimension equals transcendence degree, Transcendence degree is additive in finite towers, Relative dimension of a smooth morphism at a point, Fibres of a smooth morphism are smooth).

[L2]

$[f^{- 1} (W)]$ is the fundamental cycle of the scheme-theoretic preimage, with coefficients the generic lengths; on the reduced components $W_{j}$ with generic points $η_{j}$ the coefficient is $ℓ_{O_{X, η_{j}}} (O_{f^{- 1} (W), η_{j}})$ , and flat pullback of cycles is the linear extension of $[V] \mapsto [f^{- 1} (V)]$ (Cycles of coherent sheaves and of closed subschemes, with flat pullback, Algebraic cycles and the cycle group of a scheme of finite type over a field).

[L3]

The order function on a one-dimensional Noetherian local domain is additive, multiplicative and computed by lengths of quotients, and rational equivalence is generated by principal divisor cycles (The order function of a one-dimensional Noetherian local domain, Rational equivalence and the Chow group of cycles).

[L4]

For a flat local map $A \to B$ and a finite-length $A$ -module $M$ , a composition series of $M$ tensored with $B$ over $A$ is a filtration of $B \otimes_{A} M$ with factors $κ (p_{i}) \otimes_{A} B$ , and length is additive; if the residue-field fibre $B / m_{A} B$ has finite length $ℓ$ , the total length is $ℓ_{A} (M) \cdot ℓ_{B} (B / m_{A} B)$ when both are finite (Module length is additive in short exact sequences, Flat morphism of schemes). Finite modules over Noetherian rings have prime filtrations (Finite modules over Noetherian rings admit prime filtrations); localizing such a filtration counts minimal-prime factors by generic length, as used in step 2.1.

[L5]

Proper pushforward of cycles descends to Chow groups and if $i : Z \to X$ is a closed immersion then $i_{*}$ is the induced map on cycles (Proper pushforward of cycles and the norm formula).

Proof

technique · compute generic fibre dimensions and generic lengths for the cycle-level pullback, then descend through divisor generators and close with localization

1.1L1L2givenalgebra

Pure dimension. Let $V \subseteq Y$ be integral of dimension $d$ and put $S = f^{- 1} (V)$ . The base change $S \to V$ is flat and locally of finite type. Each generic point of an irreducible component of $S$ lies over the generic point of $V$ : on affine charts, going down makes the contraction of a minimal prime minimal, and $V$ is integral. Choose finite-type affine charts $Spec A \subseteq V$ and $Spec B \subseteq S$ meeting such a generic point, with corresponding minimal prime $p \subset B$ . Then $p \cap A = (0)$ , and $(B / p) \otimes_{A} Frac (A)$ is an integral component of the generic fibre. It has dimension $n$ by the pure-fibre hypothesis, so its function field has transcendence degree $n$ over $Frac (A)$ . The affine-domain dimension theorem and transcendence-degree additivity now give $\dim (B / p) = {trdeg}_{k} Frac (B / p) = {trdeg}_{k} Frac (A) + n = \dim A + n = d + n .$ Every nonempty affine open of an integral locally finite-type $k$ -scheme has the same function field and, by the affine-domain dimension theorem, the same dimension. The chain definition of dimension then shows the whole scheme has that dimension: any chain meets an affine neighbourhood of a point in its smallest member, where the intersections remain strict. Applying this to $V$ and to each component of $S$ identifies their global dimensions with the affine calculation above. Thus all components of $f^{- 1} (V)$ have dimension $d + n$ . If the preimage is empty its fundamental cycle is zero, which belongs to $Z_{d + n} (X)$ ; otherwise the dimension is $d + n$ . In either case its fundamental cycle is a pure $(d + n)$ -cycle.

2.1L1L2L4step 1.1algebra

Local rings over a divisor. Let $W \subseteq Y$ be integral of dimension $d + 1$ , let $r = x / y \in k (W)^{*}$ with nonzero $x, y$ in the one-dimensional local domain $A = O_{W, ζ}$ at a codimension-one point $ζ$ , and put $S = W \times_{Y} X$ . For a codimension-one point $ξ$ of $S$ over $ζ$ , set $C = O_{S, ξ}$ . The local map $A \to C$ is flat, $C$ is one-dimensional, and $x, y$ are nonzerodivisors in $C$ . Every minimal prime $p$ of $C$ contracts to $(0)$ in $A$ , so $C / p$ is a one-dimensional domain and $r$ maps to its fraction field. The generic length of the corresponding component of $S$ is $n_{p} = ℓ_{C_{p}} (C_{p})$ . A prime filtration of $C$ has $n_{p}$ factors $C / p$ for each minimal prime and only finite-length factors in addition.

3.1L3step 2.1algebra

Order calculation on the components. For a nonzerodivisor $x$ , let $χ_{x} (M) = ℓ_{C} (coker x) - ℓ_{C} (\ker x)$ . This invariant is additive on short exact sequences, vanishes on finite-length factors, and on a one-dimensional domain factor $C / p$ equals $ℓ_{C / p} ((C / p) / x (C / p))$ . Since $x, y$ are injective on $C$ , the prime filtration from step 2.1 gives $ℓ_{C} (C / x C) - ℓ_{C} (C / y C) = \sum_{p} n_{p} {ord}_{C / p} (r) .$ .

4.1L4L1L2L3step 2.1step 3.1algebra

Flat length and divisor identity. The flat local length formula [L4], applied to $A / x A$ and $A / y A$ , gives $ℓ_{C} (C / x C) - ℓ_{C} (C / y C) = {ord}_{A} (r) ℓ_{C} (C / m_{A} C) .$ The last factor is the generic length in the flat pullback of the codimension-one cycle at $ξ$ . Thus the coefficient of $f^{*} ({div}_{W} (r))$ at $ξ$ equals the coefficient of $\sum_{j} n_{j} {div}_{W_{j}} (r ∣_{W_{j}})$ . At the generic point of $W$ , $r$ is a unit and both coefficients vanish. Summing over codimension-one points and components proves $f^{*} ({div}_{W} (r)) = \sum_{j} n_{j} {div}_{W_{j}} (r ∣_{W_{j}}),$ with each component pushed to $X$ .

5.1L2L3step 4.1algebra

Descent to Chow groups. By definition ${Rat}_{d} (Y)$ is generated by the cycles $(i_{W})_{*} {div}_{W} (r)$ for integral $W \subseteq Y$ of dimension $d + 1$ and $r \in k (W)^{*}$ . The cycle-level pullback is additive, and step 4.1, applied to the closed immersion of $W$ into $Y$ and its flat base change, sends each generator to a sum of principal divisor cycles on the components of $f^{- 1} (W)$ . That sum lies in ${Rat}_{d + n} (X)$ , so $f^{*}$ descends to $A_{d} (Y) \to A_{d + n} (X)$ .

5.2L1L2step 4.1algebra

Functoriality. For the identity morphism the preimage of an integral subscheme is itself. Let $f : X \to Y$ and $g : Y \to T$ be composable flat morphisms of the stated kind. The two scheme-theoretic preimages of an integral $V \subseteq T$ agree, and the fibre dimensions add: $n_{g \circ f} = n_{g} + n_{f}$ . At a generic point of each top-dimensional component, the pullback coefficient is the length of the corresponding local tensor product; associativity of tensor products and length additivity give the same coefficient for the composite and the two successive pullbacks. Thus $(g \circ f)^{*} [V] = f^{*} g^{*} [V]$ on cycles and on Chow groups. For an open immersion $U ↪ T$ , the intersection of an integral $V \subseteq T$ with $U$ is either empty or a dense open integral subscheme of the same dimension, so pullback is restriction of cycles.

6.1L1L2L5algebra∎

Localization. Let $j : U ↪ X$ be an open immersion and $i : Z = X ∖ U ↪ X$ the complementary reduced closed subscheme. A cycle supported on $Z$ restricts to zero. Every integral closed $V \subseteq U$ is a dense open subscheme of its closure $\overline{V} \subseteq X$ , with the same function field, so $j^{*} [\overline{V}] = [V]$ and restriction is surjective. If a cycle $α$ represents a class restricting to zero, then as cycles on $U$ it is a finite (or locally finite) sum $\sum_{a} (i_{V_{a}})_{*} {div}_{V_{a}} (r_{a})$ . The functions extend to the same function fields on the closures ${\overline{V}}_{a}$ , and their divisors restrict to the stated divisors on $U$ . Hence $γ = α - \sum_{a} (i_{{\overline{V}}_{a}})_{*} {div}_{{\overline{V}}_{a}} (r_{a})$ restricts to the zero cycle on $U$ and is supported on $Z$ . The family of closures is locally finite: for every affine open $N \subseteq X$ , the open $N \cap U$ is quasi-compact because $N$ is noetherian. A locally finite family meets a quasi-compact open in only finitely many members, and if $N$ meets ${\overline{V}}_{a}$ , then the open set $N$ meets $V_{a}$ . Thus only finitely many closures meet each such $N$ . Hence $γ = i_{*} γ_{Z}$ for a (locally finite) cycle on $Z$ , proving exactness in the middle.

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

Tame symbol reciprocity in dimension two (the Key Lemma)

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Let $A$ be a two-dimensional local domain essentially of finite type over a field and $f, g \in Frac (A)^{\times}$ . For each height-one prime $q$ , define $\partial_{q} (f, g)$ by normalization and norms of the DVR symbols $(- 1)^{v (f) v (g)} f^{v (g)} / g^{v (f)}$ reduced in the residue fields. Then $\sum_{q} {ord}_{A / q} (\partial_{q} (f, g)) = 0$ . The difference between two orders of rational-section intersection is consequently a sum of divisors of these symbols; for two section zero schemes the relations can be chosen on their common intersection. This proves well-definedness of first Chern operators and Cartier Gysin on rational equivalence and commutation of two Cartier Gysins.

Facts & Assumptions

Given: the Axiom of Choice; a two-dimensional local domain $A$ , essentially of finite type over a field, with fraction field $K = Frac (A)$ , maximal ideal $m$ , and nonzero $f, g \in K^{\times}$ ; for each height-one prime $q \subseteq A$ the one-dimensional local domain $A / q$ with fraction field $κ (q)$ .

[F1]

A two-dimensional local domain essentially of finite type over a field is Noetherian, and its normalization $B$ is a finite $A$ -module and a semilocal normal domain; every local ring of a normal one-dimensional Noetherian domain at a height-one prime is a discrete valuation ring (A finite-type domain over a field has finite normalization, Height-one localizations of normal Noetherian domains are DVRs).

[F2]

The order function of a one-dimensional Noetherian local domain is multiplicative, additive and computed by lengths; it agrees with the normalized valuation on a discrete valuation ring (The order function of a one-dimensional Noetherian local domain). Length is additive in short exact sequences, and the invariant of a two-periodic complex is defined whenever its two homology modules have finite length (Module length is additive in short exact sequences).

[F3]

For a finite extension of one-dimensional local domains, the norm-order formula computes the order of a norm, with residue-field degree weights (Proper pushforward of cycles and the norm formula).

Proof

technique · direct; introduce the two-periodic invariant $e(M,a,b)$, compute it on finite-length and nilpotent modules, identify it with the order of the tame symbol on a lattice, and sum over height-one primes using normalization

1.1F2givenalgebra

The invariant and its basic properties. For a module $M$ over a commutative ring with commuting endomorphisms $a, b$ satisfying $a b M = b a M = 0$ , whose two-periodic complex has finite-length homology, set $e (M, a, b) = ℓ (\ker a / b M) - ℓ (\ker b / a M)$ , where the lengths are taken over the ring acting; both terms are finite by [F2] and the finite-homology hypothesis. If $0 \to M^{'} \to M \to M^{''} \to 0$ is a short exact sequence of such complexes, the long exact homology sequence together with additivity of length shows $e (M, a, b) = e (M^{'}, a, b) + e (M^{''}, a, b)$ . If $M$ has finite length, then $ℓ \ker a = ℓ M - ℓ (a M)$ and $ℓ \ker b = ℓ M - ℓ (b M)$ by [F2], and substitution gives $e (M, a, b) = 0$ .

2.1F2step 1.1algebra

The nilpotent identity. Let $t$ be an endomorphism of $M$ with $t^{N} M = 0$ for some $N \geq 1$ and assume $\ker (t) / t^{N - 1} M$ has finite length; consider the pair $(a, b) = (t^{i}, t^{N - i})$ for $0 \leq i \leq N$ , so that $a b = 0$ . Put $K_{i} = \ker t^{i}$ and let $a_{i} = ℓ (K_{i} / t^{N - i} M)$ , $b_{i} = ℓ (K_{i} / t K_{i + 1})$ , $c_{i} = ℓ (K_{1} / t^{i} K_{i + 1})$ , lengths over the ring acting. The quotients $K_{1} / t^{i} K_{i + 1}$ have finite length, since $t^{N - 1} M \subseteq t^{i} K_{i + 1}$ . The identities $K_{i} \cap t^{j} M = t^{j} K_{i + j}$ produce the two exact sequences $0 \to K_{1} / t^{N - i - 1} K_{N - i} \to K_{i + 1} / t^{N - i - 1} M \overset{t}{\to} K_{i} / t^{N - i} M \to K_{i} / t K_{i + 1} \to 0$ and $0 \to K_{i} / t K_{i + 1} \to K_{i + 1} / t K_{i + 2} \overset{t^{i}}{\to} K_{1} / t^{i + 1} K_{i + 2} \to K_{1} / t^{i} K_{i + 1} \to 0$ ; these sequences inductively show that all $a_{i}, b_{i}$ are finite (starting with $a_{1} = ℓ (K_{1} / t^{N - 1} M)$ ), and length additivity gives $c_{N - i - 1} - a_{i + 1} + a_{i} - b_{i} = 0$ and $b_{i} - b_{i + 1} + c_{i + 1} - c_{i} = 0$ , and with $b_{0} = c_{0} = 0$ the second relation gives $b_{i} = c_{i}$ for all $i$ , whence $a_{i + 1} - a_{i} = b_{N - i - 1} - b_{i}$ by the first; summing over $i$ yields $a_{i} = a_{N - i}$ , which is exactly $e (M, t^{i}, t^{N - i}) = 0$ .

3.1F2F3step 1.1step 2.1algebra

Multiplier identities. Let $M$ be a finite module over a Noetherian local ring, with commuting endomorphisms $a, b, x$ , $a b M = 0$ , finite-length $x$ -power torsion, and $M / x M$ supported at the closed point. Removing that torsion leaves $x$ injective, so $\ker (x a) = \ker a$ and the exact sequence $0 \to a M / x a M \to \ker b / x a M \to \ker b / a M \to 0$ gives $e (M, x a, b) = e (M, a, b) - e (a M, 0, x)$ ; the same argument gives $e (M, a, x b) = e (M, a, b) + e (b M, 0, x)$ . If $N$ is supported at the closed point and a height-one prime $q$ of a two-dimensional local domain $R$ , with $x \notin q$ , then $e_{R} (N, 0, x) = ℓ_{R_{q}} (N_{q}) {ord}_{R / q} (x) .$ Indeed, additivity and a finite filtration by powers of $q$ reduce to a finite module over the one-dimensional local domain $R / q$ ; its finite-length torsion contributes zero by step 1.1. The torsion-free quotient is a full lattice of rank $h = ℓ_{R_{q}} (N_{q})$ , and there $e (N, 0, x) = ℓ (N / x N) = {ord}_{R / q} (\det (x id)) = h {ord}_{R / q} (x)$ by the lattice-index calculation in [F3]. Now let $M$ also satisfy $M_{q} ≅ R_{q} / (π^{e + f})$ and $a = u π^{e}$ , $b = v π^{f}$ in $R_{q}$ , with $π$ a uniformizer and $u, v$ units. The nilpotent identity gives $e (M, π^{e}, π^{f}) = 0$ , while $π^{e} M$ and $π^{f} M$ have generic lengths $f$ and $e$ . If $π, u, v \in R$ , the multiplier identities yield $e_{R} (M, a, b) = - f {ord}_{R / q} (u) + e {ord}_{R / q} (v) = - {ord}_{R / q} ((- 1)^{e f} u^{f} v^{- e}) .$ For parameters initially in $R_{q}$ , first replace $π, u, v$ by $c π, c^{- e} u, c^{- f} v$ for some $c \in R ∖ q$ so $c π \in R$ , then choose $d \in R ∖ q$ with $d u, d v \in R$ and apply the formula to $d a, d b$ . The multiplier identities account for the factors $e_{R} (a M, 0, d)$ and $e_{R} (b M, 0, d)$ , whose generic lengths are $f$ and $e$ ; these are exactly the order correction from scaling the tame symbol by $d^{f - e}$ . Thus the formula also holds for the original $a, b \in R$ .

4.1F1F2step 1.1step 2.1step 3.1algebra

The normal case. Suppose first that $A$ is normal, so each height-one localization $A_{q}$ is a discrete valuation ring by [F1]. Let $q_{1}, \dots, q_{t}$ be the height-one primes containing $a b$ . Set $M = A / (a b)$ and let $M_{i}$ be the image of $M$ in $M_{q_{i}} = A_{q_{i}} / (a b)$ . The kernel and cokernel of $M \to ⨁_{i} M_{i}$ are supported only at the maximal ideal, hence have finite length. Since $a, b$ are nonzerodivisors on $A$ , cancellation gives $\ker (a : M \to M) = b M$ and $\ker (b : M \to M) = a M$ , so $e_{A} (M, a, b) = 0$ . Step 1.1 and additivity therefore give $\sum_{i} e_{A} (M_{i}, a, b) = 0$ . Write $a = u_{i} π_{i}^{e_{i}}$ and $b = v_{i} π_{i}^{f_{i}}$ in $A_{q_{i}}$ , with $π_{i}$ a uniformizer and $u_{i}, v_{i}$ units. The module $M_{i}$ has generic length $e_{i} + f_{i}$ at $q_{i}$ ; the modules $π_{i}^{e_{i}} M_{i}$ and $π_{i}^{f_{i}} M_{i}$ have generic lengths $f_{i}$ and $e_{i}$ , respectively. Here $π_{i}^{e_{i} + f_{i}}$ annihilates $M_{i}$ , and $\ker (π_{i}) / π_{i}^{e_{i} + f_{i} - 1} M_{i}$ is supported only at the maximal ideal, hence has finite length: its localization at $q_{i}$ is zero by the DVR computation and $M_{i}$ has no other height-one support. Thus the nilpotent identity of step 2.1 applies. The multiplier identities of step 3.1 then give $e_{A} (M_{i}, a, b) = - f_{i} {ord}_{A / q_{i}} (u_{i}) + e_{i} {ord}_{A / q_{i}} (v_{i}) = - {ord}_{A / q_{i}} (\partial_{A_{q_{i}}} (a, b)),$ because $\partial_{A_{q_{i}}} (a, b) = (- 1)^{e_{i} f_{i}} u_{i}^{f_{i}} / v_{i}^{e_{i}}$ in $Frac (A / q_{i})$ and the sign is a unit. Summing over $i$ proves $\sum_{q} {ord}_{A / q} (\partial_{A_{q}} (a, b)) = 0$ for normal $A$ and $a, b \in A$ .

5.1F1F3step 4.1algebra

The non-normal case. For general $A$ , let $B$ be the finite normalization of $A$ , a semilocal normal Noetherian domain by [F1], with maximal ideals $n$ and residue fields $κ (n)$ finite over $κ (m)$ . By step 4.1 applied to each local factor $B_{n}$ , $\sum_{Q \subseteq n} {ord}_{(B / Q)_{n}} (\partial_{Q} (f, g)) = 0$ for the height-one primes $Q$ of $B$ inside $n$ ; multiplying by $[κ (n) : κ (m)]$ and summing over $n$ , the norm-order formula of [F3] identifies the sum over the height-one primes $Q$ above a fixed height-one prime $q$ of $A$ with ${ord}_{A / q} (\prod_{Q ∣ q} N_{κ (Q) / κ (q)} \partial_{Q} (f, g))$ , which is the normalizing definition of $\partial_{q} (f, g)$ . Hence the weighted sum of the local identities is exactly $\sum_{q} {ord}_{A / q} (\partial_{q} (f, g)) = 0$ . Since $\partial_{q}$ and ${ord}_{A / q}$ are bimultiplicative in $f$ and $g$ , writing $f$ and $g$ as quotients of elements of $A$ extends the identity from elements to arbitrary $f, g \in K^{\times}$ .

6.1F2F3step 5.1algebra

The rational-section key formula. On an integral scheme $W$ of dimension $n$ locally of finite type over the field, let $s, t$ be nonzero rational sections of invertible sheaves $L, M$ . Choose a locally finite family of prime divisors $Z_{i}$ outside whose union both sections are generators. At the generic point of $Z_{i}$ , put $B_{i} = O_{W, Z_{i}}$ , choose local generators $s_{i}, t_{i}$ , and write $s = f_{i} s_{i}$ , $t = g_{i} t_{i}$ . Representatives of the two iterated first Chern cycles differ by the cycle $\sum_{i} ({ord}_{B_{i}} (f_{i}) {div}_{M ∣_{Z_{i}}} (t_{i} ∣_{Z_{i}}) - {ord}_{B_{i}} (g_{i}) {div}_{L ∣_{Z_{i}}} (s_{i} ∣_{Z_{i}})) = \sum_{i} {div}_{Z_{i}} (\partial_{B_{i}} (f_{i}, g_{i})),$ with all terms pushed to $W$ . To verify this equality, compare coefficients at a codimension-two point and trivialize $L, M$ there. Rescaling $s_{i}$ by a unit $u_{i}$ replaces $f_{i}$ by $u_{i}^{- 1} f_{i}$ : both sides change by $- {ord}_{B_{i}} (g_{i}) div (u_{i} ∣_{Z_{i}})$ , since the normalized DVR symbol satisfies $\partial (u_{i}^{- 1}, g_{i}) = {\overset{ˉ}{u}}_{i}^{- {ord}_{B_{i}} (g_{i})}$ ; norms give the same identity for nonnormal $B_{i}$ by the finite norm-order computation in Proper pushforward of cycles and the norm formula. The analogous rescaling of $t_{i}$ also preserves the equality. We may therefore take the generators to be these trivializations, when the left side is zero and the right coefficient is zero by step 5.1. This proves the key formula without restricting a section that vanishes identically on $Z_{i}$ .

7.1step 6.1F2F3algebra

First Chern descent and commutation. The right side of the key formula is a locally finite sum of principal divisors, so the two iterated first Chern classes on $[W]$ agree in $A_{n - 2} (W)$ . Taking $M = O_{W}$ and $t = r \in k (W)^{*}$ gives $c_{1} (L) \cap {div}_{W} (r) = c_{1} (O_{W}) \cap {div}_{L} (s) = 0$ : the section $1$ of the trivial bundle has zero divisor on every integral cycle. Thus first Chern operators annihilate each rational-equivalence generator and commute on Chow groups. Pushforward along integral closed subschemes and linear extension give the same conclusions on arbitrary locally finite type schemes.

8.1F3step 6.1step 7.1algebra

Cartier Gysin descent with support. Let $D = Z (s)$ for a global section of $L$ , and test on an integral $W$ . If $W \subseteq D$ , its Gysin is the first Chern operator on $W$ , which descends by step 7.1. Otherwise $D ∣_{W}$ is Cartier. For a rational section $t$ of $M ∣_{W}$ , apply step 6.1, including the components of $D ∣_{W}$ among the $Z_{i}$ , and choose $s_{i} = s ∣_{W}$ when $Z_{i} \subseteq̸ D$ . Then $f_{i} = 1$ for these indices, so their tame symbols are $1$ . The remaining principal divisors are on $Z_{i} \subseteq D$ , and therefore give a rational equivalence on $D \cap W$ , proving $D^{!} {div}_{M ∣_{W}} (t) = c_{1} (M ∣_{D \cap W}) \cap [D \cap W] .$ For $M = O_{W}$ and $t = r$ the right side is zero. Hence Cartier Gysin annihilates principal-divisor relations in its target Chow group, including after base change when the pulled-back section is not Cartier. Proper compatibility used to push these computations to the ambient zero scheme is the norm-order formula in [F3] when a cycle is not contained in $D$ , and the same formula applied to rational sections when it is contained.

9.1step 6.1step 8.1algebra∎

Two Cartier Gysins. For zero schemes $D = Z (s)$ and $D^{'} = Z (t)$ , choose rational sections on $W$ equal to the given sections whenever $W$ is not contained in their zero scheme; if it is contained, choose any nonzero rational section of the corresponding restricted line bundle. Choose local generators $s_{i} = s$ outside $D$ and $t_{i} = t$ outside $D^{'}$ . The two sides of the key formula represent the two iterated Gysins. Its right side has no term outside $D \cap D^{'}$ , because there $f_{i} = 1$ or $g_{i} = 1$ . Thus the principal-divisor relations occur on subvarieties of $D \cap D^{'} \cap W$ , proving commutation in the required target Chow group, including cycles contained in either divisor.

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

Proper pushforward commutes with flat pullback

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Let $k$ be a field and let $\begin{array}{ccc} X^{'} & \overset{g^{'}}{\to} & X \\ ↓ f^{'} & ↓ f \\ Y^{'} & \overset{g}{\to} & Y \end{array}$ be a cartesian square of schemes locally of finite type over $k$ (Fibre product of schemes), with $f$ proper (hence $f^{'}$ proper, Properness survives arbitrary base change) and $g$ flat of relative dimension $n$ with all fibres of pure dimension $n$ (hence $g^{'}$ flat of relative dimension $n$ , Flatness is stable under arbitrary base change). Then the two graded homomorphisms $A_{d} (X) \to A_{d + n} (Y^{'})$ agree: $g^{*} \circ f_{*} = f_{*}^{'} \circ g^{' *},$ where $f_{*}, f_{*}^{'}$ are proper pushforwards (Proper pushforward of cycles and the norm formula) and $g^{*}, g^{' *}$ are flat pullbacks (Flat pullback of cycles and of rational equivalence).

Facts & Assumptions

Given: the Axiom of Choice; a cartesian square as in the statement, with $f$ proper and $g$ flat of pure relative dimension $n$ ; a coherent $O_{X}$ -module $F$ supported in dimension at most $d$ .

[L1]

The cycle of a coherent sheaf $[F]_{d}$ is the sum of the generic lengths over the $d$ -dimensional components of the support, additive in short exact sequences with a common dimension bound, and flat pullback of cycles is the linear extension of $[V] \mapsto [f^{- 1} V]$ (Cycles of coherent sheaves and of closed subschemes, with flat pullback).

[L2]

Proper pushforward of cycles is the norm-degree pushforward on integral cycles and descends to Chow groups; flat pullback of cycles is defined for flat morphisms of pure relative dimension $n$ and descends to Chow groups (Proper pushforward of cycles and the norm formula, Flat pullback of cycles and of rational equivalence).

[L3]

A proper quasi-finite morphism is finite; on affine schemes the global sections of a quasi-coherent sheaf are computed by the Čech complex of a finite affine cover, and flat base change of tensor products commutes with kernels (A proper quasi-finite morphism is finite, Affine quasi-coherent sheaves are modules, Cech cohomology computes quasi-coherent cohomology on a separated scheme). Length is additive in short exact sequences (Module length is additive in short exact sequences).

Proof

technique · direct; compare the two operations on the cycles of coherent sheaves by computing generic lengths, then apply the flat-base-change isomorphism for the direct image sheaf

1.1L1L2L3givenalgebra

Proper pushforward of the cycle of a sheaf. Let $F$ be coherent on $X$ with support of dimension at most $d$ . Then $f_{*} [F]_{d} = [f_{*} F]_{d}$ in $Z_{d} (Y)$ . Indeed, let $η$ be the generic point of a $d$ -dimensional component of $Supp (f_{*} F)$ ; every point of $Supp F$ mapping to $η$ is generic in a $d$ -dimensional support component, because otherwise its closure would have dimension less than $d$ and map onto a neighbourhood of $η$ of dimension $d$ , which is impossible; hence the fibre of the support over $η$ is finite. Give that support the closed scheme structure defined by $Ann F$ ; $F$ is a coherent sheaf on this closed subscheme. Its restricted morphism to $Y$ is proper and is quasi-finite over $η$ , so after removing the closed image of its non-quasi-finite locus it is finite near $η$ by [L3]. Then $(f_{*} F)_{η} ≅ ⨁_{ξ \mapsto η} F_{ξ}$ is a finite module of finite length over the local ring $O_{Y, η}$ , and a composition series over each local ring gives $ℓ_{O_{Y, η}} ((f_{*} F)_{η}) = \sum_{ξ \mapsto η} [κ (ξ) : κ (η)] ℓ_{O_{X, ξ}} (F_{ξ})$ , with the residue degree weights because the simple factors of $F_{ξ}$ acquire composition factors of $κ (ξ)$ over $κ (η)$ ; comparing with the norm-degree definition of $f_{*} [F]_{d}$ proves the identity. If the image of a component has dimension less than $d$ , both $d$ -cycles vanish at that component.

1.2L1L2givenalgebra

Flat pullback of the cycle of a sheaf. For $F$ coherent on $Y$ supported in dimension at most $d$ one has $g^{*} [F]_{d} = [g^{*} F]_{d + n}$ in $Z_{d + n} (Y^{'})$ : at a generic point $ξ$ of a top-dimensional component of the preimage of its support, with $η = g (ξ)$ , the stalk of $g^{*} F$ is $F_{η} \otimes_{O_{Y, η}} O_{Y^{'}, ξ}$ . Tensoring a composition series of the finite-length module $F_{η}$ with this flat local algebra shows that its length is $ℓ_{O_{Y, η}} (F_{η}) ℓ_{O_{Y^{'}, ξ}} (O_{Y^{'}, ξ} / m_{η} O_{Y^{'}, ξ})$ . The second factor is the generic fibre multiplicity and can exceed one for a nonreduced fibre.

1.3L3givenalgebra

Flat base change for the direct image. For a quasi-coherent $O_{X}$ -module $G$ there is a natural isomorphism $g^{*} f_{*} G ≅ f_{*}^{'} g^{' *} G$ : on an affine open $Spec A \subseteq Y^{'}$ whose image in $Y$ is contained in an affine open over which $f$ is proper and $G$ is quasi-coherent, the preimage under $f$ has a finite affine cover with affine finite intersections (properness gives separatedness and quasi-compactness), the Čech complex computes the degree-zero sections by [L3], tensoring with $A$ over the base ring is exact, and the comparison maps agree on overlaps; hence they glue.

2.1L1L2step 1.1step 1.2step 1.3algebra∎

Compatibility on cycles. Apply steps 1.1, 1.2 and 1.3 to $F = O_{V}$ for an integral closed subscheme $V \subseteq X$ of dimension $d$ : $g^{*} f_{*} [V] = g^{*} [f_{*} O_{V}]_{d} = [g^{*} f_{*} O_{V}]_{d + n} = [f_{*}^{'} g^{' *} O_{V}]_{d + n} = f_{*}^{'} [g^{' *} O_{V}]_{d + n} = f_{*}^{'} g^{' *} [V]$ , where the middle equality is step 1.3 applied to the quasi-coherent sheaf $O_{V}$ and the outer equalities are steps 1.1 and 1.2; the finite-support and locally finite cases are handled by the same identities term by term. Since both sides are additive, $g^{*} f_{*} = f_{*}^{'} g^{' *}$ on cycles and therefore on Chow groups by [L2].

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Intersection with an invertible sheaf and the first Chern class

Definition

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Let $k$ be a field and let $X$ be a scheme locally of finite type over $k$ . Let $L$ be an invertible $O_{X}$ -module (Invertible sheaves).

Integral case. Let $W \subseteq X$ be integral of dimension $n$ , with function field $k (W)$ (Sheaf total quotient rings, The field of fractions $Frac (D) = (D ∖ {0})^{- 1} D$ of an integral domain), and let $s$ be a nonzero rational section of $L ∣_{W}$ (Rational section line bundle). The Weil divisor of $s$ is ${div}_{L} (s) = \sum_{Z \subseteq W} {ord}_{O_{W, Z}} (s) [Z] \in Z_{n - 1} (W),$ the sum over the codimension-one integral closed subschemes $Z \subseteq W$ , with the order function of The order function of a one-dimensional Noetherian local domain read on a local generator of $L$ at the generic point of $Z$ ; the sum is locally finite, and finite if $W$ is of finite type. The first Chern class $c_{1} (L) \cap [W] : = [{div}_{L} (s)] \in A_{n - 1} (W)$ is independent of the chosen nonzero rational section (two such differ by a rational function, whose divisor is rationally equivalent to zero), and therefore defines a codimension-one Chow class; on a smooth $W$ this is the first Chern class in its Chow ring.
General case. For an integral closed subscheme $i : W ↪ X$ of dimension $d + 1$ set $c_{1} (L) \cap [W] : = i_{*} (c_{1} (i^{*} L) \cap [W]) \in A_{d} (X)$ , where $i_{*}$ is the proper pushforward (Proper pushforward of cycles and the norm formula); extend $Z$ -linearly. The resulting operation $c_{1} (L) \cap - : A_{d + 1} (X) \to A_{d} (X)$ is well defined on Chow groups (this is the content of the next two references) and graded.

Basic properties. (i) $c_{1} (L \otimes_{O_{X}} N) \cap α = (c_{1} (L) + c_{1} (N)) \cap α$ and $c_{1} (O_{X}) \cap α = 0$ for all $α$ . (ii) If $X$ is pure of dimension $d + 1$ and a global section $s$ of $L$ is a nonzerodivisor on $O_{X}$ , its zero scheme $D$ is an effective Cartier divisor and $c_{1} (L) \cap [X] = [D]$ in $A_{d} (X)$ . More generally the same formula on a pure-dimensional closed subscheme $Y$ requires $s ∣_{Y}$ to be a nonzerodivisor on $O_{Y}$ . (iii) $c_{1} (L) \cap -$ commutes with proper pushforward and flat pullback: for $f : X \to Y$ proper, $f_{*} (c_{1} (f^{*} L) \cap α) = c_{1} (L) \cap f_{*} α$ , and for $f$ flat of relative dimension $n$ , $f^{*} (c_{1} (L) \cap α) = c_{1} (f^{*} L) \cap f^{*} α$ . (iv) $c_{1} (L)$ depends only on the isomorphism class of $L$ .

Cartier Gysin. For a section $s$ of $L$ with zero scheme $j : D ↪ X$ , after any base change define $j^{!} : A_{m} (X^{'}) \to A_{m - 1} (D^{'})$ on integral $V \subset X^{'}$ by the Cartier divisor of $s ∣_{V}$ if $V$ is not contained in $D^{'}$ , and by $c_{1} (L ∣_{V}) \cap [V]$ if it is; push this class from $D^{'} \cap V$ to $D^{'}$ . This works even when the pulled-back zero divisor is not Cartier. It commutes with proper pushforward and flat pullback, two Cartier Gysins commute, and $j^{!} j_{*} β = c_{1} (L ∣_{D}) \cap β$ .

Well-definedness. All the constructions above are local in the integral cycle, so they are defined for locally finite cycles as well. In the integral case, a nonzero rational section $s$ of $L ∣_{W}$ is a rational multiple of a local generator, so the order function of The order function of a one-dimensional Noetherian local domain is defined at the generic point of every codimension-one integral closed subscheme $Z \subseteq W$ , and only finitely many $Z$ meeting any fixed affine chart receive a nonzero order: on that chart $W$ is a Noetherian domain and $s$ is represented by a fraction whose numerator and denominator vanish on only finitely many height-one primes, exactly as in the finiteness discussion of Rational equivalence and the Chow group of cycles. The class $[{div}_{L} (s)]$ is independent of $s$ because the ratio of two rational sections is a rational function of $k (W)^{*}$ and principal divisors lie in ${Rat}_{n - 1} (W)$ ; this is the definition of rational equivalence. In the general case the operation is extended by proper pushforward along $i$ and by linearity; that it descends through rational equivalence on $X$ is the content of Tame symbol reciprocity in dimension two (the Key Lemma) (the tame symbol reciprocity that controls the difference of two iterated Cartier intersections) together with the facts that proper pushforward and flat pullback each descend to Chow groups (Flat pullback of cycles and of rational equivalence, Proper pushforward of cycles and the norm formula). Property (ii) is the cycle computation of Cycles of coherent sheaves and of closed subschemes, with flat pullback. For (iii), the flat-pullback identity is the generic-length calculation in Flat pullback of cycles and of rational equivalence applied to the divisor cycle; for proper pushforward, trivializing the line bundle at codimension-one generic points reduces the identity to the norm-order formula in Proper pushforward of cycles and the norm formula. Both identities are checked on integral cycles and extend linearly. Property (iv) follows because an isomorphism of invertible sheaves identifies their local generators and hence the resulting divisor coefficients.

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

Localization sequence for Chow groups and homotopy invariance of affine space

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. For a closed immersion $j : Z ↪ T$ with complement $u : U ↪ T$ , $A_{m} (Z) \overset{j_{*}}{\to} A_{m} (T) \overset{u^{*}}{\to} A_{m} (U) \to 0$ is exact. Projection $T \times A^{r} \to T$ gives an isomorphism $A_{m} (T) ≅ A_{m + r} (T \times A^{r})$ . These statements hold for finite type schemes over a field and locally finite cycles on locally finite type schemes.

Facts & Assumptions

Given: the Axiom of Choice; a closed immersion $j : Z ↪ T$ with open complement $u : U ↪ T$ ; a field $k$ over which $T$ is locally of finite type; the projection $π : T \times A^{r} \to T$ .

[L1]

Cycles, rational equivalence and the Chow group are as in Rational equivalence and the Chow group of cycles and Algebraic cycles and the cycle group of a scheme of finite type over a field; the order function defines divisors of rational functions on integral subschemes (The order function of a one-dimensional Noetherian local domain).

[L2]

The closed immersion $j$ is proper, so $j_{*}$ is defined on cycles and descends to Chow groups; the open immersion $u$ is flat of relative dimension $0$ , so $u^{*}$ is restriction of cycles (Proper pushforward of cycles and the norm formula, Flat pullback of cycles and of rational equivalence).

[L3]

The projection $π$ is flat of relative dimension $r$ with fibres $A^{r}$ , so $π^{*} : A_{m} (T) \to A_{m + r} (T \times A^{r})$ is defined (Flat pullback of cycles and of rational equivalence).

[L4]

The two-dimensional tame-symbol reciprocity identity makes Cartier Gysin along a regular Cartier divisor well defined on Chow groups; in particular, intersection with $t = 0$ annihilates rational-equivalence generators (Tame symbol reciprocity in dimension two (the Key Lemma)).

Proof

technique · lift cycles by closure for localization; for affine-space homotopy, use minimal polynomials for surjectivity and Cartier Gysin of the zero section for a left inverse

1.1L1L2givenalgebra

Localization. The composite $u^{*} j_{*}$ is zero because a cycle supported on $Z$ restricts to zero on $U$ . Every integral closed $V \subseteq U$ is a dense open subscheme of its closure $\overline{V} \subseteq T$ , with the same function field, so $u^{*} [\overline{V}] = [V]$ and $u^{*}$ is surjective. If a cycle $α$ represents a class whose restriction to $A_{*} (U)$ is zero, then as cycles on $U$ it is a finite (or locally finite) sum $u^{*} α = \sum_{a} (i_{V_{a}})_{*} {div}_{V_{a}} (r_{a}) .$ The functions extend to the same function fields on the closures ${\overline{V}}_{a}$ , and their divisors restrict to the displayed divisors on $U$ . Hence $γ = α - \sum_{a} (i_{{\overline{V}}_{a}})_{*} {div}_{{\overline{V}}_{a}} (r_{a})$ restricts to the zero cycle on $U$ and is supported on $Z$ . The family of closures is locally finite: for every affine open $N \subseteq T$ , the open $N \cap U$ is quasi-compact because $N$ is noetherian; a locally finite family meets a quasi-compact open in only finitely many members, and if $N$ meets ${\overline{V}}_{a}$ , then openness of $N$ implies it meets $V_{a}$ . Thus only finitely many closures meet each such $N$ , and $γ = j_{*} γ_{Z}$ for a (locally finite) cycle $γ_{Z}$ on $Z$ . This proves exactness in the middle, also for locally finite cycles.

2.1L1L3step 1.1algebra

Surjectivity for $A^{1}$ . Let $V \subseteq T \times A^{1}$ be integral of dimension $m + 1$ , let $W = \overline{π (V)}$ , and write $t$ for the coordinate. Since the fibres of $π$ have dimension one, $\dim W$ is either $m$ or $m + 1$ . If $\dim W = m$ , the generic fibre of $V \to W$ is a closed integral subscheme of dimension one in $A_{k (W)}^{1}$ , hence is the whole affine line; because $V$ is closed and has the same dimension as $W \times A^{1}$ , it follows that $V = W \times A^{1}$ and $[V] = π^{*} [W]$ . If $\dim W = m + 1$ , the generic fibre is a closed point of $A_{k (W)}^{1}$ , cut out by its monic irreducible polynomial $P (t) \in k (W) [t]$ . After shrinking to a dense open $W^{\circ} \subseteq W$ , the coefficients of $P$ are regular and the ideal of $V \cap (W^{\circ} \times A^{1})$ is generated by $P$ : equality with this principal ideal holds over the generic point and spreads after shrinking because the ideals are finitely generated. The monic equation makes this zero scheme finite flat over $W^{\circ}$ , so it has no vertical codimension-one components; its generic fiber is integral, hence its only component is $V ∣_{W^{\circ}}$ with multiplicity one. Thus $[V ∣_{W^{\circ}}]$ is the principal divisor of $P$ and is rationally equivalent to zero on $W^{\circ} \times A^{1}$ . By localization, $[V]$ is rationally equivalent to a cycle supported over $W ∖ W^{\circ}$ . Each integral component $C$ of that cycle has dimension $m + 1$ and image closure $W^{'} \subseteq W ∖ W^{\circ}$ of dimension at most $m$ . The fibres of $C \to W^{'}$ have dimension at most one, so $\dim W^{'} = m$ and the generic fibre is the whole affine line; since $C$ is closed in the integral scheme $W^{'} \times A^{1}$ and has its full dimension, $C = W^{'} \times A^{1}$ . Each resulting cycle is therefore a pullback $π^{*} [W^{'}]$ . For a locally finite family of input components $V$ , the closures $W$ are locally finite: if a quasi-compact open $N \subseteq T$ meets $W = \overline{π (V)}$ , then $V$ meets $π^{- 1} (N)$ , a quasi-compact open, so only finitely many input components contribute. Each individual divisor and closure construction is locally finite, so the resulting pullback and residual cycles are locally finite. For the residual components $C = W^{'} \times A^{1}$ , local finiteness also follows because $W^{'}$ meets $N$ exactly when $C$ meets the zero-section copy of $N$ . Iterating over the $r$ coordinates gives surjectivity for $A^{r}$ .

3.1L1L3L4step 2.1algebra∎

Injectivity for $A^{1}$ . Let $σ : T \to T \times A^{1}$ be the zero section and $D = T \times {0}$ . For an integral $(m + 1)$ -dimensional $V \subseteq T \times A^{1}$ not contained in $D$ , define $σ^{!} [V]$ to be the pushforward to $T$ of ${div}_{V} (t)$ ; this divisor is supported on $V \cap D$ . If $V \subseteq D$ , set $σ^{!} [V] = 0$ : this is the Cartier Gysin value because the normal line of $D$ is trivial and the first Chern class of the trivial line bundle is zero. Extend additively to cycles. The two-dimensional tame-symbol reciprocity in [L4] says that Cartier Gysin along $D$ sends each principal-divisor relation on an integral $(m + 2)$ -dimensional subscheme to a rationally trivial $m$ -cycle: its local terms are divisors of the tame symbols on the curve components of the intersection with $D$ . Thus $σ^{!}$ annihilates rational equivalence and descends to Chow groups. For an integral $V \subseteq T$ , the coordinate is a nonzerodivisor on $V \times A^{1}$ and ${div}_{V \times A^{1}} (t) = [V \times {0}],$ so $σ^{!} π^{*} [V] = [V]$ . Therefore $σ^{!}$ is a left inverse of $π^{*}$ and $π^{*}$ is injective. Together with step 2.1, this proves $A_{m} (T) ≅ A_{m + 1} (T \times A^{1})$ ; iteration over the $r$ coordinates gives $A_{m} (T) ≅ A_{m + r} (T \times A^{r})$ .

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Chow groups of projective space

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Let $k$ be a field and $n \geq 0$ . Then for every $d$ : $A_{d} (P_{k}^{n}) = {\begin{cases} Z \cdot [P^{d}], & 0 \leq d \leq n, \\ 0, & otherwise, \end{cases}$ where $P^{d} \subseteq P^{n}$ is a $d$ -dimensional linear subspace and $[P^{d}]$ its class (Rational equivalence and the Chow group of cycles, Algebraic cycles and the cycle group of a scheme of finite type over a field). In particular the degree homomorphism $\deg : A_{0} (P_{k}^{n}) \overset{≅}{\to} Z, \sum_{x} m_{x} [x] ⟼ \sum_{x} m_{x} [κ (x) : k],$ is an isomorphism; its inverse sends $1$ to $[P^{0}]$ for any $k$ -rational point, and every $0$ -cycle whose support consists of $k$ -rational points has class $(\sum_{x} m_{x}) [P^{0}]$ . For $k = C$ this is the classical cellular decomposition. More generally, $A_{d} (A_{k}^{n}) = Z$ for $d = n$ and $A_{d} (A_{k}^{n}) = 0$ otherwise: homotopy invariance shifts degrees by $n$ , and $A_{*} (Spec k)$ is $Z$ in degree $0$ , so every cycle on affine space of dimension less than $n$ is rationally equivalent to zero. For $n = 1$ , a closed point of degree greater than one is the principal divisor of its monic irreducible polynomial; linear polynomials suffice for rational points.

Facts & Assumptions

Given: the Axiom of Choice; a field $k$ and $n \geq 0$ ; the linear subspaces $P^{d} \subseteq P^{n}$ .

[L1]

The localization sequence $A_{m} (Z) \overset{i_{*}}{\to} A_{m} (T) \overset{j^{*}}{\to} A_{m} (U) \to 0$ is exact for a closed immersion $i : Z ↪ T$ with open complement $U$ , and the projection $T \times A^{r} \to T$ induces $A_{m} (T) ≅ A_{m + r} (T \times A^{r})$ (Localization sequence for Chow groups and homotopy invariance of affine space).

[L2]

First Chern classes of invertible sheaves define graded cap operations $c_{1} (L) \cap - : A_{m} (X) \to A_{m - 1} (X)$ which are additive and commute with proper pushforward and flat pullback; on an integral $W$ with a rational section $s$ not vanishing identically, $c_{1} (L) \cap [W] = [{div}_{L} (s)]$ (Intersection with an invertible sheaf and the first Chern class).

[L3]

Proper pushforward of a closed point $x$ to $Spec k$ is $[κ (x) : k]$ times the fundamental class of the point, by the norm-degree definition (Proper pushforward of cycles and the norm formula, Rational equivalence and the Chow group of cycles).

[L4]

The standard affine charts of $P_{k}^{n}$ are affine $n$ -space, and $P_{k}^{n} ∖ H ≅ A_{k}^{n}$ for a coordinate hyperplane $H$ (Relative projective space from standard charts, Projective space is Proj of a polynomial ring, Twisting sheaf on Proj).

Proof

technique · direct; use localization and homotopy invariance for the affine and projective decompositions, and the cap operation of $\mathcal O(1)$ followed by degree for independence

1.1L1L4givenalgebra

Affine space. Applying homotopy invariance [L1] successively in the $n$ coordinates gives $A_{d} (A_{k}^{n}) ≅ A_{d - n} (Spec k)$ , which is $Z$ for $d = n$ and $0$ otherwise; the top class is $[A^{n}]$ and for $n = 1$ every closed point is the principal divisor of its monic irreducible polynomial in $k [t]$ , which is linear precisely for a rational point.

2.1L1step 1.1givenalgebra

Generation of the Chow groups of projective space. For $n = 0$ , projective space is $Spec k$ and the assertion is immediate. Assume $n \geq 1$ . Let $H = P^{n - 1} \subseteq P^{n}$ be a coordinate hyperplane with complement $A^{n}$ . Localization [L1] gives the exact sequence $A_{d} (P^{n - 1}) \overset{i_{*}}{\to} A_{d} (P^{n}) \overset{j^{*}}{\to} A_{d} (A^{n}) \to 0$ . For $d < n$ the group $A_{d} (A^{n})$ vanishes by step 1.1, so $i_{*}$ is surjective, and induction on $n$ proves that $A_{d} (P^{n})$ is generated by the class $[P^{d}]$ of a $d$ -dimensional linear subspace: in the hyperplane the class of a $d$ -dimensional linear subspace generates by induction, and its pushforward is the class of the corresponding linear subspace of $P^{n}$ ; the induction starts at $n = d$ , where $P^{d}$ has no hyperplane below it. For $d = n$ the same exact sequence reads $0 = A_{n} (P^{n - 1}) \to A_{n} (P^{n}) \to A_{n} (A^{n}) ≅ Z \to 0$ by step 1.1, so $A_{n} (P^{n}) = Z [P^{n}]$ is generated by the top class, and the cycle group $Z_{n} (P^{n}) = Z [P^{n}]$ admits no nonzero rational equivalences because none of its subvarieties has dimension $n + 1$ . For $d < 0$ or $d > n$ there are no integral closed subschemes of dimension $d$ , so $A_{d} (P^{n}) = 0$ .

3.1L2L3step 2.1algebra

Independence. Fix $d$ with $0 \leq d \leq n$ and let $c_{1} = c_{1} (O (1)) \cap -$ ; by [L2] the iterated cap $c_{1}^{d}$ maps $A_{m} (P^{n})$ to $A_{m - d} (P^{n})$ , is additive, and is defined by cutting with $d$ coordinate hyperplanes in general position. On the linear subspace $P^{d}$ , the $d$ coordinate hyperplanes cut it in a single $k$ -rational point, so $c_{1}^{d} [P^{d}] = [P^{0}]$ , and the degree homomorphism $\deg : A_{0} (P^{n}) \to Z$ of [L3] sends $[P^{0}]$ to $1$ ; hence $\deg \circ c_{1}^{d}$ is a homomorphism $A_{d} (P^{n}) \to Z$ sending the generator $[P^{d}]$ of step 2.1 to $1$ . A cyclic group admitting a homomorphism onto $Z$ with generator mapping to $1$ is infinite cyclic, so $A_{d} (P^{n}) = Z \cdot [P^{d}]$ .

4.1L3step 3.1givenalgebra∎

The degree isomorphism and closed points. By steps 2.1 and 3.1, $A_{0} (P^{n})$ is generated by $[P^{0}]$ for a $k$ -rational point; for a closed point $x$ with residue field $κ (x)$ , proper pushforward to $Spec k$ is multiplication by $[κ (x) : k]$ by [L3], so $[x] = [κ (x) : k] [P^{0}]$ and $\deg$ is the stated isomorphism; a $0$ -cycle supported on $k$ -rational points has class $(\sum_{x} m_{x}) [P^{0}]$ . The computation over $C$ is the classical cellular decomposition under the same identification.

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

Deformation to the normal cone and specialization

Definition

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth normal-sequence suppliers. For a closed immersion $i : X ↪ Y$ with ideal $I$ , put $C_{X} Y = {Spec}_{X} (⨁_{n \geq 0} I^{n} / I^{n + 1})$ and $M = {Bl}_{X \times {\infty}} (Y \times P^{1})$ . Let $B = {Bl}_{X} Y$ be the strict transform of $Y \times {\infty}$ . The fibre at infinity is the sum $M_{\infty} = B + P (C_{X} Y \oplus 1)$ of effective Cartier divisors, whose intersection is $P (C_{X} Y)$ ; this is a union with a common boundary, not a disjoint union. Projectivized cones here use the lines convention $Proj ({gr}_{I} O_{Y} [S])$ . Off infinity $M ≅ Y \times A^{1}$ , and $D_{i} : = M ∖ B$ has special fibre $C_{X} Y$ and ordinary fibres $Y$ . It is flat over $P^{1}$ . The specialization $σ_{i} : A_{m} (Y) \to A_{m} (C_{X} Y)$ is obtained by extending the flat pullback of a cycle to $D_{i}$ , then taking Cartier Gysin at infinity. It sends an integral $[V]$ to the fundamental cycle $[C_{X \cap V} V]$ pushed to $C_{X} Y$ . For a regular immersion of codimension $d$ , the normal cone equals the rank- $d$ normal bundle $N_{X / Y}$ . The construction works for finite type schemes over a field, with locally finite cycles in the locally finite type case.

Well-definedness. The charts of $M$ near infinity are computed directly: over an affine open $Spec A \subseteq Y$ with $X = Spec (A / I)$ and $t$ the coordinate vanishing at infinity, the $t$ -chart of the blowup is the affine blowup algebra $A [t, I / t] = \sum_{n \geq 0} t^{- n} I^{n} [t] \subseteq A [t, t^{- 1}]$ , and multiplication by $t$ is injective with quotient ${gr}_{I} A$ ; in the chart of a generator $a \in I$ the ring is $A [I / a] [t / a]$ , and $t = a (t / a)$ exhibits the fibre at $t = 0$ as the sum of the exceptional divisor $a = 0$ and the strict transform $t / a = 0$ , whose charts are ${Bl}_{X} Y$ and whose intersection is $Proj {gr}_{I} A$ . This identifies the fibre at infinity as the stated union of effective Cartier divisors, without assuming $Y$ integral or the centre Cartier, and shows that, away from infinity, $M ∖ B ≅ Y \times A^{1}$ , while its special fibre is $C_{X} Y$ ; the charts are torsion-free over $k [t]$ (or polynomial over the affine blowup algebras), and a torsion-free module over the principal ideal domain $k [t]$ is flat, giving flatness of $M$ over $P^{1}$ (compare the chart computation of Smooth immersions, their conormal sequence, deformation charts and smooth sections). Localization for the pair $C_{X} Y ↪ M ∖ B$ gives a lift of the pulled-back cycle class to $D_{i}$ , and two lifts differ by a class supported on $C_{X} Y$ ; the Cartier Gysin at infinity kills that difference because the normal line of the infinity fibre is trivial, so $j^{!} j_{*} γ = c_{1} (O) \cap γ = 0$ by the basic property $c_{1} (O_{X}) = 0$ of Intersection with an invertible sheaf and the first Chern class; hence $σ_{i}$ is well defined on Chow classes. On an integral cycle $[V]$ the closure of $V \times A^{1}$ in the $t$ -chart is $Spec O_{V} [t, I O_{V} / t]$ , and cutting by $t$ produces ${gr}_{I O_{V}} O_{V}$ , which is the fundamental cycle of the normal cone $C_{X \cap V} V$ with its generic lengths; for a regular sequence generating $I$ , Associated graded algebra of an ideal generated by a regular sequence identifies ${gr}_{I} A$ with ${Sym}_{A / I} (I / I^{2})$ , so the cone is the rank- $d$ normal bundle of Smooth immersions, their conormal sequence, deformation charts and smooth sections.

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

The projective bundle formula for Chow groups

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Let $k$ be a field, let $X$ be a scheme locally of finite type over $k$ , and let $E$ be a finite locally free $O_{X}$ -module of rank $r \geq 1$ (Locally free sheaves of finite rank). Let $π : P (E) = {Proj}_{X} Sym (E) ⟶ X$ be the projective bundle in the quotient convention, with tautological quotient $π^{*} E \to O_{P (E)} (1)$ (Projective bundle in the quotient convention, Relative Proj of a graded quasi-coherent algebra, Projective bundle represents line quotients), and put $ξ : = c_{1} (O (1))$ (Intersection with an invertible sheaf and the first Chern class); $π$ is smooth of relative dimension $r - 1$ (Relative dimension of a smooth morphism at a point), so flat pullback $π^{*}$ is defined (Flat pullback of cycles and of rational equivalence, Fibres of a smooth morphism are smooth). Then for every $d \in Z$ the map $⨁_{i = 0}^{r - 1} A_{d + i} (X) ⟶ A_{d + r - 1} (P (E)), (α_{0}, \dots, α_{r - 1}) ⟼ \sum_{i = 0}^{r - 1} ξ^{i} \cap π^{*} α_{i},$ is an isomorphism of abelian groups. More precisely, $π_{*} (ξ^{s} \cap π^{*} α) = 0$ for $s < r - 1$ and $π_{*} (ξ^{r - 1} \cap π^{*} α) = α$ for all $α \in A_{*} (X)$ (Proper pushforward of cycles and the norm formula).

No smoothness of $X$ is assumed; if $X$ is smooth equidimensional of dimension $n$ then $P (E)$ is smooth equidimensional and the formula is a statement about the codimension-graded Chow groups $A^{*} (X)$ , $A^{*} (P (E))$ .

Facts & Assumptions

Given: the Axiom of Choice; a field $k$ ; a scheme $X$ locally of finite type over $k$ ; a finite locally free $O_{X}$ -module $E$ of rank $r \geq 1$ ; the projective bundle $π : P (E) \to X$ with tautological quotient $O (1)$ and $ξ = c_{1} (O (1))$ .

[L1]

$π$ is flat, projective (hence proper) and smooth of relative dimension $r - 1$ , with fibres $P^{r - 1}$ ; over an open $U \subseteq X$ on which $E$ is trivial, $P (E) ∣_{U} ≅ U \times P^{r - 1}$ compatibly with $O (1)$ (Projective bundle in the quotient convention, Projective bundle represents line quotients, Relative dimension of a smooth morphism at a point).

[L2]

The first Chern class $ξ$ defines a graded cap operation $ξ \cap - : A_{m} \to A_{m - 1}$ which is additive, commutes with proper pushforward and flat pullback, and satisfies $ξ \cap ξ^{s} \cap - = ξ^{s + 1} \cap -$ (Intersection with an invertible sheaf and the first Chern class).

[L3]

The localization sequence and affine homotopy invariance hold, and the Chow groups of projective space over a field are $Z$ in each dimension $0, \dots, n$ generated by the linear classes (Localization sequence for Chow groups and homotopy invariance of affine space, Chow groups of projective space).

[L4]

Proper pushforward of an integral cycle $W$ with $\dim π (W) < \dim W$ is zero, and pullback of classes along the flat morphism $π$ shifts degree by $r - 1$ (Proper pushforward of cycles and the norm formula, Flat pullback of cycles and of rational equivalence, Relative projective space from standard charts).

Proof

technique · direct; compute the pushforwards of the powers of $\xi$ on integral classes, then derive injectivity from the resulting triangular identities and surjectivity from a trivializing open with Noetherian induction

1.1L1L2L3L4givenalgebra

Pushforward identities. Let $V \subseteq X$ be an integral closed subscheme of dimension $m$ , so that $π^{*} [V] = [P (E ∣_{V})]$ is an $(m + r - 1)$ -cycle. First suppose $E ∣_{V}$ is trivial. The zero schemes of $r - 1$ coordinate sections of $O (1)$ are relative hyperplanes meeting in a section $σ : V \to P (E ∣_{V})$ , so $ξ^{r - 1} \cap π^{*} [V] = σ_{*} [V]$ and $π_{*} σ_{*} [V] = [V]$ . For any $E$ and $0 \leq s < r - 1$ , the cycle $ξ^{s} \cap π^{*} [V]$ has dimension $m + r - 1 - s > m$ and is supported on $P (E ∣_{V})$ , whose image has dimension $m$ ; hence its components push to zero by [L4]. For the top power with general $E$ , choose a dense open $U \subseteq V$ trivializing the bundle. The identity $π_{*} (ξ^{r - 1} \cap π^{*} [V]) = [V]$ holds over $U$ , and its difference is in $A_{m} (V)$ supported on $V ∖ U$ . This complement has dimension less than $m$ , so localization [L3] makes the difference zero. Both identities extend by linearity to all $α \in A_{*} (X)$ .

1.2L1L2L3givenalgebra

Surjectivity over a trivializing open. Let $U \subseteq X$ be an open on which $E$ is trivial, so that $P (E) ∣_{U} ≅ U \times P^{r - 1}$ . The Chow group of $U \times P^{r - 1}$ is generated by the classes $ξ^{i} \cap π^{*} α$ with $0 \leq i \leq r - 1$ and $α \in A_{*} (U)$ : stratify $P^{r - 1}$ by the affine spaces $A^{j}$ , use the localization sequence and affine homotopy invariance [L3] to reduce every class to classes pulled back from the strata, and represent each stratum class as an $i$ -fold cut with $ξ$ by repeated Cartier cutting of coordinate hyperplanes, which realizes it as $ξ^{i} \cap π^{*}$ of a class on $U$ ; the claim is compatible with the restriction to any smaller open by [L2].

2.1L2step 1.1algebra

Injectivity. Suppose $\sum_{i = 0}^{r - 1} ξ^{i} \cap π^{*} α_{i} = 0$ . Applying $π_{*}$ and step 1.1 gives $α_{r - 1} = 0$ , since the other summands push to zero. Capping the remaining relation with $ξ^{j}$ and applying $π_{*}$ successively gives $α_{r - 1 - j} = 0$ for every $0 \leq j \leq r - 2$ by the same computation, using associativity of repeated caps from [L2]. Hence the displayed map is injective.

2.2L2L3step 1.2givenalgebra

Global surjectivity. It suffices to express an integral cycle $[V]$ on $P (E)$ , and we induct on the dimension of the reduced image closure $W \subseteq X$ . The cycle is supported on $P (E ∣_{W})$ . Choose a dense open $U \subseteq W$ trivializing the bundle. Step 1.2 expresses $[V] ∣_{U}$ in the required form; lift its coefficient classes from $A_{*} (U)$ to $A_{*} (W)$ using localization [L3]. The resulting difference is the pushforward of a cycle class on $P (E ∣_{W ∖ U})$ . Each integral component of that cycle has image closure of dimension strictly less than $\dim W$ , so induction expresses its class in the required form over its image. Push these expressions to $W$ and then $X$ , using compatibility of flat pullback with a closed immersion and of caps with proper pushforward. The induction terminates because the integral image has finite dimension. For locally finite cycles the constructions remain locally finite: $π$ is proper, so the image closures of a locally finite family are locally finite; each residual family is supported within its respective image closure, and the dimension bound for a fixed-degree input cycle is uniform. Thus the same argument sums locally finitely without asserting that the whole base has finitely many components.

3.1L1step 2.1step 2.2given∎

Conclusion. Injectivity is step 2.1 and surjectivity is step 2.2; both hold degreewise, so the displayed map is an isomorphism for every $d$ . The final paragraph of the statement follows because $E$ locally free of rank $r$ makes $π$ smooth of relative dimension $r - 1$ by [L1], and smoothness of $X$ inherits to the total space of a smooth morphism.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Bivariant Chow operations and bivariant classes

Definition

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Fix a field $k$ and work in the category of schemes locally of finite type over $k$ , with $k$ -morphisms. For $f : Z \to T$ in this category, a degree- $d$ bivariant Chow operation assigns to every $k$ -morphism $T^{'} \to T$ in this category homomorphisms $c : A_{m} (T^{'}) \to A_{m - d} (Z \times_{T} T^{'})$ , commuting with proper pushforward, flat pullback of fixed relative dimension, and Cartier divisor Gysin. Restriction along any base morphism, composition, and proper pushforward of such operations are defined by their action. Flat pullback and Cartier divisor Gysin give bivariant classes. If $h : Z \to Z_{1}$ is proper over $T$ , pushing a class for $Z \to T$ along $h$ gives a class for $Z_{1} \to T$ ; ordinary proper pushforward is a covariant Chow operation, not a class reversing that proper morphism. Equality can be tested on fundamental classes of integral schemes over $T$ ; it suffices to test after a proper birational modification of each such integral scheme.

Well-definedness. The three axioms are required for every base change $T^{'} \to T$ in this category, and the operations are compared only on Chow groups, using Rational equivalence and the Chow group of cycles. Proper pushforward and flat pullback are the maps of Proper pushforward commutes with flat pullback; the Cartier divisor Gysin is the operation of Intersection with an invertible sheaf and the first Chern class. Restriction along a base morphism $T^{''} \to T^{'}$ simply changes the indexing family, and composition of operations is associative because each axiom is applied first to the inner and then to the outer factor; the compatibility of two Cartier Gysins is the tame-symbol reciprocity computation of Tame symbol reciprocity in dimension two (the Key Lemma), and if a cycle is contained in one of the two divisors the operation is $c_{1}$ of the corresponding invertible sheaf, whose commutation is the same rational-section argument. For a proper $h : Z \to Z_{1}$ over $T$ the definition $(h_{*} c)_{T^{'}} = h_{T^{'}, *} c_{T^{'}}$ satisfies the three axioms by the corresponding compatibilities of proper pushforward with flat pullback and Cartier Gysin. For the equality criterion, a finite cycle is a finite sum of pushforwards of integral fundamental classes. For a locally finite cycle $\sum_{i} n_{i} [V_{i}]$ on $T^{'}$ , the map $h : ∐_{i} V_{i} \to T^{'}$ is proper: over every quasi-compact open only finitely many supports occur, and their inclusions are closed immersions. Locally finite Chow groups on this disjoint union are the product of the Chow groups of its components, as are those of its base change to $Z$ . Flat restriction to each open-and-closed component therefore determines an operation on the class $(n_{i} [V_{i}])_{i}$ . Equality on the integral classes gives equality there, and proper compatibility pushes it along $h$ to equality on the original cycle. Finally, for a proper birational $h : V^{'} \to V$ of integral schemes one has $h_{*} [V^{'}] = [V]$ , so equality on $[V^{'}]$ pushes forward to equality on $[V]$ . This is a descent test for operations, not an assertion that Chow pullback to a modification is injective.

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

Operational Chern classes and the Whitney formula

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Fix a field $k$ and work with schemes locally of finite type over $k$ and $k$ -morphisms. Every rank- $r$ vector bundle $E$ on such a scheme $T$ has operators $c_{j} (E) : A_{m} (T^{'}) \to A_{m - j} (T^{'})$ for every $T^{'} \to T$ , defined by the projective bundle relation and commuting with the bivariant operations. They satisfy $c_{0} = 1$ , $c_{j} = 0$ for $j > r$ , $c (L) = 1 + c_{1} (L)$ , arbitrary base restriction, $c (E) = c (E^{'}) c (E^{''})$ for $0 \to E^{'} \to E \to E^{''} \to 0$ , and a splitting principle by iterated projective bundles whose flat pullback is injective after every base change. If a section of $E$ on a pure $n$ dimensional scheme $T$ has zero scheme regularly embedded of codimension $r$ , then $c_{r} (E) \cap [T] = [Z (s)]$ in $A_{n - r} (T)$ . These are operators on singular schemes; a Chow ring is not required.

Facts & Assumptions

Given: the Axiom of Choice; a field $k$ ; a scheme $T$ locally of finite type over $k$ ; a rank- $r$ vector bundle $E$ on $T$ ; for every base change $T^{'} \to T$ the projective bundle $π : P (E_{T^{'}}) \to T^{'}$ with tautological quotient $O (1)$ and $ξ = c_{1} (O (1))$ .

[L1]

Projective bundle formula: for every $d$ the map $⨁_{i = 0}^{r - 1} A_{d + i} (T^{'}) \to A_{d + r - 1} (P (E_{T^{'}}))$ , $(α_{i}) \mapsto \sum_{i} ξ^{i} \cap π^{*} α_{i}$ , is an isomorphism, and $π_{*} (ξ^{r - 1} \cap π^{*} α) = α$ while $π_{*} (ξ^{s} \cap π^{*} α) = 0$ for $s < r - 1$ (The projective bundle formula for Chow groups).

[L2]

Bivariant operations and their compatibilities; the first Chern class cap operation $ξ \cap -$ commutes with proper pushforward and flat pullback, and two Cartier operations commute (Bivariant Chow operations and bivariant classes, Intersection with an invertible sheaf and the first Chern class).

[L3]

Proper pushforward and flat pullback of the relevant degrees, and the fact that a line bundle $L$ has $P (L) ≅ T^{'}$ with $ξ = c_{1} (L)$ (Proper pushforward of cycles and the norm formula, Flat pullback of cycles and of rational equivalence, Projective bundle in the quotient convention, Locally free sheaves of finite rank).

Proof

technique · direct; define the classes by the projective bundle relation, verify the axioms by uniqueness, then run the Whitney and section arguments by induction on the rank

1.1L1L3givenalgebra

Definition by the projective bundle relation. For rank zero set $c_{0} = 1$ and all positive Chern operators to zero; the Whitney and section assertions in this case are identities, and no projective bundle of rank zero is used. Assume now $r \geq 1$ . For every base change $T^{'} \to T$ and every class $α \in A_{m} (T^{'})$ , expand $ξ^{r} \cap π^{*} α = \sum_{j = 0}^{r - 1} ξ^{j} \cap π^{*} β_{j}$ with $β_{j} \in A_{m - r + j} (T^{'})$ , which exists and is unique by [L1], and define $c_{i} (E) \cap α : = (- 1)^{i + 1} β_{r - i}$ for $1 \leq i \leq r$ ; explicitly the defining relation is $\sum_{j = 0}^{r} (- 1)^{j} ξ^{r - j} \cap π^{*} (c_{j} (E) \cap α) = 0$ , with $c_{0} = 1$ and $c_{j} = 0$ for $j > r$ by convention. All terms have the same dimension, so the relation determines the $c_{j} (E) \cap α$ uniquely by the basis part of [L1]. For a line bundle $L$ , $P (L) = T^{'}$ and the relation reads $ξ \cap α - c_{1} (L) \cap α = 0$ , giving $c (L) = 1 + c_{1} (L)$ by [L3].

2.1L2step 1.1algebra

Compatibility with bivariant operations. Let $b$ be any bivariant operation commuting with $π^{*}$ and $ξ$ (in the sense $b ξ = ξ b$ ). Applying $b$ to the defining relation of step 1.1 and using the commutation with $π^{*}$ and $ξ$ gives the same relation with $b (c_{j} (E) \cap α)$ in place of $c_{j} (E) \cap α$ ; by the uniqueness in step 1.1, $b c_{j} (E) = c_{j} (E) b$ . Taking for $b$ a proper pushforward, a flat pullback or a Cartier operation gives the compatibility axioms of the operational classes by [L2]. Restriction along an arbitrary base morphism merely restricts the indexing family of operations: the defining projective-bundle relation is the same relation on each further base scheme. No flatness of that base morphism is needed.

2.2L1step 1.1algebra

The Whitney formula and splitting principle. Iterating projective bundles of successive kernels of tautological line quotients gives a flag tower on which any vector bundle has a filtration with line-bundle quotients. Every projection has injective flat pullback by [L1], with left inverse given by its top relative hyperplane cap followed by pushforward; the same holds after every base change. First let $E$ already have a filtration with line quotients $L_{1}, \dots, L_{r}$ in subbundle order. In the quotient convention the inclusion $L_{1} ↪ E$ induces a section of $O (1) \otimes π^{*} L_{1}^{- 1}$ on $P (E)$ whose zero divisor is $P (E / L_{1})$ ; in a local splitting it is a coordinate hyperplane, so it is Cartier even over a singular base. Repeat on this divisor with the next line subbundle of the quotient, ending with the empty projective bundle. Cartier cutting and commutation of first Chern operations therefore give $\prod_{i = 1}^{r} (ξ - π^{*} c_{1} (L_{i})) \cap π^{*} α = 0$ for every $α$ , where each notation $π^{*} c_{1} (L_{i})$ means the cap of the pulled-back line bundle. Expanding and using uniqueness in step 1.1 shows $c (E) = \prod_{i} (1 + c_{1} (L_{i}))$ as operations. For $0 \to E^{'} \to E \to E^{''} \to 0$ , pull back to the combined flag towers of $E^{'}$ and $E^{''}$ . Their line filtrations concatenate to a filtration of the pulled-back $E$ , by taking inverse images of the filtration of $E^{''}$ . The product formula just proved then gives $c (E) = c (E^{'}) c (E^{''})$ upstairs; injectivity of the tower pullback, after every further base change, descends this operator identity. These same towers prove the stated splitting principle.

3.1L1L2step 1.1step 2.2algebra∎

The section formula. Assume $s$ a section of $E$ with $Z (s) \subseteq T$ regularly embedded of codimension $r$ , $T$ pure of dimension $n$ . Induct on $r$ , the case $r = 1$ being the Cartier divisor formula $c_{1} (E) \cap [T] = [Z (s)]$ of [L2] with $E$ a line bundle. For $r > 1$ , let $t$ be the image of $s$ under $π^{*} E \to O (1)$ on $P (E)$ and $H = \ker (π^{*} E \to O (1))$ ; the zero scheme of the section $t$ of $O (1)$ is a divisor, and on it the section lifts to $H$ with zero scheme $π^{- 1} Z (s)$ . At every point of $π^{- 1} Z (s)$ the local regular sequence of $r$ equations for $Z (s)$ is transformed by an invertible change of generators to the equation $t$ together with $r - 1$ equations on the fibre direction, so it remains regular; hence $π^{- 1} Z (s)$ is regularly embedded of codimension $r - 1$ in $Z (t)$ , and $π^{- 1} Z (s)$ is the zero scheme of the restricted section of $H$ on the Cartier divisor $Z (t)$ . The induction hypothesis applied on $Z (t)$ gives $c_{r - 1} (H) \cap [Z (t)] = [π^{- 1} Z (s)]$ , while the Cartier formula for $Z (t)$ and the Whitney formula of step 2.2 give $c_{r} (π^{*} E) \cap [P (E)] = c_{r - 1} (H) \cap c_{1} (O (1)) \cap [P (E)] = c_{r - 1} (H) \cap [Z (t)]$ ; combining identifies the flat pullbacks of the two required classes; injectivity of $π^{*}$ from [L1] then yields $c_{r} (E) \cap [T] = [Z (s)]$ .

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

Homotopy invariance for vector bundles

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Fix a field $k$ and let $T$ be locally of finite type over $k$ . For any rank- $r$ vector bundle $p : E \to T$ , flat pullback $p^{*} : A_{m} (T) \to A_{m + r} (E)$ is bijective, including after every $k$ -base change $T^{'} \to T$ with $T^{'}$ locally of finite type over $k$ . Its inverse is denoted $s_{E}^{!}$ .

Facts & Assumptions

Given: the Axiom of Choice; a field $k$ ; a scheme $T$ locally of finite type over $k$ and a rank- $r$ vector bundle $p : E \to T$ .

[L1]

The projective bundle $q : P (E^{\lor} \oplus 1) \to T$ in the quotient convention has tautological quotient $O (1)$ with $ξ = c_{1} (O (1))$ ; the complement of the infinity divisor $j : P (E^{\lor}) ↪ P (E^{\lor} \oplus 1)$ , cut out by the section of $O (1)$ coming from the trivial summand, is canonically $E$ , and $q ∣_{E} = p$ (The projective bundle formula for Chow groups, Intersection with an invertible sheaf and the first Chern class).

[L2]

Localization sequence and affine-space homotopy invariance (Localization sequence for Chow groups and homotopy invariance of affine space).

[L3]

Projective bundle formula on $P (E^{\lor})$ and $P (E^{\lor} \oplus 1)$ : the classes $1, ξ, \dots$ form a basis over the corresponding Chow groups, and caps by $ξ$ commute with proper pushforward (The projective bundle formula for Chow groups, Intersection with an invertible sheaf and the first Chern class).

Proof

technique · direct; compactify the bundle by the projective completion, identify the image of the infinity pushforward inside the projective bundle basis by means of the trivial-summand section, and read off the quotient

1.1L1L2givenalgebra

Compactification and localization. If $r = 0$ , $E = T$ and $p = id$ , so pullback and its inverse are the identity after every base change. Assume $r \geq 1$ for the compactification argument. Let $q : P (E^{\lor} \oplus 1) \to T$ be the projective completion with $O (1)$ and $ξ = c_{1} (O (1))$ , and let $j : P (E^{\lor}) ↪ P (E^{\lor} \oplus 1)$ be the infinity divisor, the zero scheme of the section of $O (1)$ induced by the direct-summand $1 \subseteq E^{\lor} \oplus 1$ . Its complement is $E$ , and the restriction of $q$ to $E$ is $p$ ; hence the localization sequence of [L2] gives the exact sequence $A_{m} (P (E^{\lor})) \overset{j_{*}}{\to} A_{m} (P (E^{\lor} \oplus 1)) \overset{}{\to} A_{m} (E) \to 0$ .

2.1L3step 1.1algebra

The image of the infinity pushforward. Write $π = q \circ j$ . For an integral cycle $[V]$ on $T$ , the trivial-summand section cuts the relative hyperplane $P (E^{\lor} ∣_{V})$ in the projective completion, with multiplicity one. The Cartier formula therefore gives $j_{*} π^{*} [V] = ξ \cap q^{*} [V]$ , and linearity gives this identity for all classes $β$ on $T$ . Compatibility of the first Chern cap with proper pushforward then yields $j_{*} (ξ^{a} \cap π^{*} β) = ξ^{a + 1} \cap q^{*} β$ . Consequently, on the basis $1, ξ, \dots, ξ^{r - 1}$ over $A_{*} (T)$ of the source and $1, ξ, \dots, ξ^{r}$ of the target supplied by [L3], the image of $j_{*}$ is exactly the span of the positive powers $ξ^{1}, \dots, ξ^{r}$ .

3.1L1L3step 1.1step 2.1algebra∎

Conclusion. By step 2.1 the quotient of $A_{*} (P (E^{\lor} \oplus 1))$ by the image of $j_{*}$ is the direct summand $q^{*} A_{*} (T)$ spanned by $1$ , and by step 1.1 this quotient is exactly $A_{*} (E)$ ; since $q ∣_{E} = p$ , the induced map is the flat pullback $p^{*}$ , which is therefore an isomorphism onto the summand spanned by the classes $q^{*} α$ with $m$ shifted by $r$ . Both bundles and both bases pull back along any $k$ -base change $T^{'} \to T$ with $T^{'}$ locally of finite type over $k$ , so the same computation applies verbatim after base change; the inverse of $p^{*}$ is by definition $s_{E}^{!}$ .

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

Koszul resolutions and restriction to flat fibres

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. A section $s$ of a rank- $d$ bundle $Q$ is a regular section here if, at each point of its zero scheme $Z (s)$ , some local frame expresses $s$ by $d$ components forming a regular sequence of length $d$ in the local ring. Under this hypothesis it has exact Koszul resolution $⋀^{d} Q^{\lor} \to \dots \to Q^{\lor} \to O \to O_{Z (s)} \to 0$ . Tensoring with a vector bundle preserves it. If a coherent sheaf $H$ has a finite locally free resolution on a scheme and $t$ is a Cartier equation acting injectively on $H$ , restricting the resolution to $t = 0$ remains exact and resolves $H / t H$ . In the deformation blowup for a closed embedding of smooth quasi-projective varieties $i : X ↪ Y$ , let $E$ be a finite locally free sheaf on $X$ . The strict transform $F : X \times P^{1} ↪ M$ is disjoint from $B = {Bl}_{X} Y$ , and a resolution of $F_{*} {pr}_{X}^{*} E$ restricts exactly to the ordinary $Y$ fibre and to $P = P_{l i n e s} (N \oplus 1)$ ; on $B$ it is an exact complex.

Facts & Assumptions

Given: the Axiom of Choice; a rank- $d$ vector bundle $Q$ on a scheme with a section $s$ whose $d$ local components form a regular sequence of length $d$ at every point of $Z (s)$ ; a coherent sheaf $H$ with a finite locally free resolution and a Cartier equation $t$ acting injectively on $H$ ; a closed embedding $i : X ↪ Y$ of smooth quasi-projective varieties and a finite locally free sheaf $E$ on $X$ ; the deformation blowup $M = {Bl}_{X \times {\infty}} (Y \times P^{1})$ of Deformation to the normal cone and specialization.

[L1]

A sequence $a_{1}, \dots, a_{d}$ is regular on a module $M$ if each $a_{i}$ is a nonzerodivisor on $M / (a_{1}, \dots, a_{i - 1}) M$ ; the Koszul complex of a regular sequence has zero positive homology, and its mapping-cone construction inducts on $d$ (Regular Sequences Give Acyclic Koszul Complexes). Away from $Z (s)$ some component is a unit in the local ring.

[L2]

On a regular quasi-projective scheme of finite type over a field, every coherent sheaf has a finite locally free resolution, and tensoring with a vector bundle preserves exactness of complexes of vector bundles (Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes, Locally free sheaves of finite rank).

[L3]

The deformation blowup $M$ is smooth and quasi-projective with strict transform $F$ isomorphic to $X \times P^{1}$ ; it is flat over $P^{1}$ with ordinary fibres $Y$ ; the open $M ∖ B$ has special fibre $C_{X} Y$ , while the complete fibre at infinity is the union $B + P (N \oplus 1)$ with intersection $P (N)$ (Deformation to the normal cone and specialization, Smooth immersions, their conormal sequence, deformation charts and smooth sections).

Proof

technique · direct; prove Koszul exactness by the mapping-cone induction, derive fibre restriction from the vanishing of Tor against a principal quotient, and apply both to the deformation blowup

1.1L1L2givenalgebra

Koszul exactness. At a point of $Z (s)$ choose the frame in the hypothesis and let $K (a_{1}, \dots, a_{d}; M)$ be the Koszul complex. For $d = 0$ it is $M$ in degree zero. For $d > 0$ , by [L1] it is the mapping cone of multiplication by $a_{d}$ from $K (a_{1}, \dots, a_{d - 1}; M)$ to itself, so its long exact homology sequence identifies $H_{i} (K (a_{1}, \dots, a_{d}))$ with the kernel and cokernel of the map induced by $a_{d}$ on $H_{*} (K (a_{1}, \dots, a_{d - 1}))$ ; by induction $a_{d}$ acts on the only nonzero homology $H_{0} = M / (a_{1}, \dots, a_{d - 1}) M$ as a nonzerodivisor, so the only nonzero homology of the full complex is $H_{0} = M / (a_{1}, \dots, a_{d}) M$ and the complex resolves that quotient. At a point outside $Z (s)$ a component $a_{j}$ is a unit; if $e_{j}$ is the corresponding exterior generator, the homotopy $h (v) = a_{j}^{- 1} e_{j} \land v$ satisfies $\partial h + h \partial = id$ by the contraction differential, so the complex is exact there and resolves the zero stalk. The construction is canonical under change of frame, because exterior powers and contraction by $s$ are, so the local complexes glue to the global resolution $⋀^{d} Q^{\lor} \to \dots \to Q^{\lor} \to O \to O_{Z (s)} \to 0$ ; tensoring with a vector bundle $W$ preserves exactness and produces the resolution of $W ∣_{Z (s)}$ twisted by the Koszul terms.

1.2L2givenalgebra

Restriction to a flat fibre. Let $0 \to H_{n} \to \dots \to H_{0} \to H \to 0$ be a finite locally free resolution and let $t$ be a Cartier equation acting injectively on $H$ ; since the $H_{i}$ are locally free, $t$ acts injectively on each of them. Tensoring with the two-term resolution $0 \to O \overset{t}{\to} O \to O / (t) \to 0$ of the principal quotient and computing Tor, the identity ${Tor}_{1} (H, O / (t)) = \ker (t : H \to H) = 0$ together with the snake lemma applied to the short exact sequences of complexes shows that ${Tor}_{i} (H, O / (t)) = 0$ for all $i > 0$ , and the complex $H_{∙} / t H_{∙}$ is exact with $H_{0} (H_{∙} / t H_{∙}) = H / t H$ . Hence the restricted complex $H_{∙} ∣_{t = 0}$ remains exact and resolves $H / t H$ ; the essential extra hypothesis is injectivity on $H$ : injectivity on the locally free terms alone does not suffice.

2.1L2L3step 1.1step 1.2algebra∎

The deformation blowup. By [L3], $M$ is smooth and quasi-projective over the field, so every coherent sheaf on $M$ has a finite locally free resolution by [L2]; the strict transform $F$ is isomorphic to $X \times P^{1}$ and meets the fibre at infinity in the section $[0 : 1]$ of $P = P (N \oplus 1)$ , while it is disjoint from $B = {Bl}_{X} Y$ , since $B$ meets $P$ in $P (N)$ and the strict transform of $X \times P^{1}$ lies over $X \times {\infty}$ inside the exceptional component. On the chart where the infinity fibre is Cartier with equation $t$ , the ordinary fibre $Y$ has equation $t - 1$ , and both act injectively on the coherent sheaf $F_{*} {pr}_{X}^{*} E$ , since on its support $F ≅ X \times P^{1}$ they are parameter equations on the second factor and ${pr}_{X}^{*} E$ is locally free there. This sheaf need not be locally free on $M$ . Near its intersection with $P$ the component $B$ is absent, so the Cartier equation of $P$ agrees with that of the infinity fibre; step 1.2 therefore restricts a finite locally free resolution of $F_{*} {pr}_{X}^{*} E$ exactly to the ordinary fibre and to $P$ , where the restricted complex resolves the restrictions of the sheaf. Near $B$ the sheaf is zero, and a bounded exact complex of vector bundles resolving the zero sheaf is split exact by induction on its length, starting from the last cokernel; hence the restriction of the resolution to $B$ is an exact complex, as asserted.

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

Zero-section Gysin and excess intersection for vector subbundles

Statement

All schemes and base changes below are locally of finite type over a fixed field $k$ , and all morphisms are $k$ -morphisms.

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. For a rank- $r$ vector bundle $p : N \to T$ , let $s_{N}^{!} = (p^{*})^{- 1}$ . If $0 \to N^{'} \to N \to Q \to 0$ is exact and $a : N^{'} ↪ N$ is the vector subbundle immersion, then $s_{N}^{!} a_{*} p^{' *} α = c_{rank Q} (Q) \cap α$ . In particular $s_{N}^{!} s_{*} α = c_{r} (N) \cap α$ . These identities and proper/flat/divisor compatibility hold after every base change.

Facts & Assumptions

Given: the Axiom of Choice; a rank- $r$ vector bundle $p : N \to T$ ; an exact sequence $0 \to N^{'} \to N \to Q \to 0$ of vector bundles with subbundle immersion $a : N^{'} ↪ N$ and projection $p^{'} : N^{'} \to T$ .

[L1]

Flat pullback along a vector bundle is bijective with inverse $s_{N}^{!}$ , compatibly with every base change (Homotopy invariance for vector bundles).

[L2]

Chern classes are operational, commute with bivariant operations, satisfy $c_{r} (N) \cap [T] = [Z (s)]$ when $T$ is pure-dimensional and a section of $N$ has regularly embedded zero scheme of codimension $r$ , and obey the Whitney formula (Operational Chern classes and the Whitney formula, Bivariant Chow operations and bivariant classes).

Proof

technique · direct; pull the identity back along the bijection $p^*$, where it becomes the regular-section formula for the universal quotient section

1.1L1L2givenalgebra

Compatibility. Since $p^{*}$ is bijective with inverse $s_{N}^{!}$ by [L1], the displayed identities are equivalent after applying $p^{*}$ to the corresponding identities of bivariant operations: proper, flat and divisor compatibility of $s_{N}^{!}$ follows by applying $p^{*}$ and the corresponding push-pull or Cartier identities to each equality; all constructions are stable under base change by [L1] and [L2].

1.2L2givenalgebra

The excess formula. Consider the universal section of $p^{*} Q$ on $N$ , namely the image of the vector coordinate under the composite $N \to Q$ (equivalently the section of $p^{*} Q$ whose value at a point is the class of the tautological vector); its zero scheme is exactly the subbundle $N^{'}$ . After a local splitting of $0 \to N^{'} \to N \to Q \to 0$ the section cuts $rank Q$ independent fibre coordinates, so its zero scheme is regularly embedded of codimension $rank Q$ . For an integral $V \subseteq T$ , write $N_{V} = p^{- 1} (V)$ and $N_{V}^{'} = p^{' - 1} (V)$ . These are integral and pure-dimensional, of dimensions $\dim V + r$ and $\dim V + rank N^{'}$ , with flat-pullback cycles $[N_{V}] = p^{*} [V]$ and $[N_{V}^{'}] = p^{' *} [V]$ . On $N_{V}$ the restricted universal section again cuts independent fibre coordinates, so [L2] applies there. Pushing its section formula along $N_{V} ↪ N$ and using proper compatibility and flat naturality of Chern operators gives $a_{*} p^{' *} [V] = c_{rank Q} (p^{*} Q) \cap p^{*} [V] = p^{*} (c_{rank Q} (Q) \cap [V])$ . Applying $s_{N}^{!}$ and extending linearly gives $s_{N}^{!} a_{*} p^{' *} α = c_{rank Q} (Q) \cap α$ for all $α$ .

2.1L2step 1.2givenalgebra∎

The zero-section case. Taking $N^{'} = 0$ and $Q = N$ gives the zero section $s : T \to N$ of $N$ and the identity $s_{N}^{!} s_{*} α = c_{r} (N) \cap α$ ; the local splittings used in step 1.2 verify regularity only and do not assert a global splitting of the sequence.

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

Relative projective bundles: K-theory generation by tautological twists

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. Let $Y$ be smooth quasi-projective over a field and $E$ a rank- $r$ bundle. For $q : P = P_{l i n e s} (E) \to Y$ , $K^{0} (P) = G_{0} (P)$ is generated as a $K^{0} (Y)$ -module by $O (a)$ , $0 \leq a < r$ . For $r \geq 1$ and $a \geq 0$ , $q_{!} O (a) = {Sym}^{a} (E^{\lor})$ . For $r = 0$ , the projective bundle is empty, its $K$ -groups and all $q_{!} O (a)$ are zero, and the assertion of generation is vacuous. This statement has the corresponding dual form for quotient projective bundles.

Facts & Assumptions

Given: the Axiom of Choice; a smooth quasi-projective $k$ -scheme $Y$ ; a rank- $r$ vector bundle $E$ on $Y$ ; the projective bundle $q : P = P_{l i n e s} (E) = P_{q u o t} (E^{\lor}) \to Y$ with tautological line $L = O (- 1)$ , quotient bundle $Q$ of rank $r - 1$ and universal sequence $0 \to O (- 1) \to q^{*} E \to Q \to 0$ .

[L1]

On $P$ , smooth quasi-projective over a field, $K^{0} (P) = K_{0} (P)$ and every coherent sheaf has a finite locally free resolution (Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes).

[L2]

The relative diagonal of $P \times_{Y} P$ is the zero scheme of the section $L_{1} \to Q_{2}$ of $L_{1}^{\lor} \otimes Q_{2}$ , whose $r - 1$ components near the diagonal are a regular sequence of fibre-coordinate differences, with exact Koszul resolution of terms ${pr}_{1}^{*} O (- j) \otimes {pr}_{2}^{*} ⋀^{j} Q^{\lor}$ ; the resolution restricts exactly to fibres and survives base change (Koszul resolutions and restriction to flat fibres, Projective bundle in the quotient convention).

[L3]

The pushforward $q_{!}$ of Pushforward of coherent sheaves in algebraic K-theory satisfies the projection formula, flat base change preserves its values, and the Čech complex of the standard projective affine cover computes the relevant cohomology (Projection formula for higher direct images and K-theory pushforward, Cech cohomology computes quasi-coherent cohomology on a separated scheme).

Proof

technique · direct; push the diagonal Koszul resolution forward to obtain the twist identity, express the exterior factors through the universal sequence, and read off generation and the pushforward of the twists

1.1L1L2L3givenalgebra

The diagonal identity. If $r = 0$ , $P = \emptyset$ by the projective-bundle definition, so every class and pushforward is zero and the assertion is immediate; assume $r \geq 1$ from here onward. The section of [L2] meets the stronger regular-section hypothesis: trivialize $E$ over $Spec A$ and, near a diagonal point, use the same projective chart on both factors with lines generated by $e_{0} + \sum_{i = 1}^{r - 1} u_{i} e_{i}$ and $e_{0} + \sum_{i = 1}^{r - 1} v_{i} e_{i}$ . Modulo the second line, the first generator has components $u_{i} - v_{i}$ in the quotient basis $e_{1}, \dots, e_{r - 1}$ over $A [u_{1}, \dots, u_{r - 1}, v_{1}, \dots, v_{r - 1}]$ . Successively eliminating $u_{i}$ makes each next difference monic in a new variable, hence a nonzerodivisor over any base ring; these $r - 1$ components form a regular sequence, and the argument survives every base change on $A$ . Off the diagonal some component is a unit locally, because the two residue-field lines are distinct. For $r = 1$ the sequence is empty and the relative diagonal is the whole product. Thus [L2] supplies the exact Koszul resolution in all cases $r \geq 1$ . For a vector bundle $H$ on $P$ , tensor the Koszul resolution of the diagonal of [L2] with ${pr}_{1}^{*} H$ and push forward along ${pr}_{2}$ : the diagonal term contributes $H$ , because ${pr}_{2}$ restricts to the identity on the diagonal, and the $j$ -th Koszul term contributes $q^{*} (q_{!} (H (- j))) \otimes [⋀^{j} Q^{\lor}]$ by the projection formula and flat base change [L3]. Hence $[H] = \sum_{j = 0}^{r - 1} (- 1)^{j} q^{*} (q_{!} (H (- j))) [⋀^{j} Q^{\lor}]$ in $K^{0} (P)$ .

2.1L1step 1.1algebra

Generation. The dual universal sequence $0 \to Q^{\lor} \to q^{*} E^{\lor} \to O (1) \to 0$ gives the exterior-power identity $[⋀^{j} q^{*} E^{\lor}] = [⋀^{j} Q^{\lor}] + [⋀^{j - 1} Q^{\lor}] [O (1)]$ , by the two graded pieces of the exterior-power filtration in a local splitting. Solving this recurrence, starting with $⋀^{0} Q^{\lor} = O$ , gives $[⋀^{j} Q^{\lor}] = \sum_{a = 0}^{j} (- 1)^{a} [q^{*} ⋀^{j - a} E^{\lor}] [O (a)]$ ; substituting into step 1.1 shows that every class $[H]$ of a vector bundle is a finite combination of $q^{*} K^{0} (Y)$ -classes tensored with $O (a)$ , $0 \leq a < r$ . By [L1] every coherent sheaf is resolved by vector bundles, so the same classes generate $K_{0} (P)$ ; the action of $q^{*} K^{0} (Y)$ is the module structure and this proves generation.

3.1L3givenalgebra∎

Pushforward of the twists. For $a \geq 0$ , the sheaf $q_{!} O (a)$ is computed on the standard projective affine cover [L3]: the local projective-space cohomology calculation has only degree-zero cohomology with its homogeneous monomial basis of degree $a$ , and changes of trivialization act on this basis by ${Sym}^{a} (E^{\lor})$ ; higher direct images vanish by the same computation and the Čech cover comparison. Hence $q_{!} O (a) = {Sym}^{a} (E^{\lor})$ as a vector bundle on $Y$ , which is the second assertion; the dual statement for quotient projective bundles follows by dualizing $E$ and using $P (E^{\lor})$ with the induced universal sequence.

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

Gysin specialization is bivariant and compatible with base change

Statement

All schemes and base changes below are locally of finite type over a fixed field $k$ , and all morphisms are $k$ -morphisms.

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth-immersion and homological suppliers. For a closed immersion $Z ↪ T$ with a surjection $N^{\lor} ↠ I / I^{2}$ from a rank- $d$ bundle on $Z$ , each $T^{'} \to T$ gives $Z^{'} = Z \times_{T} T^{'}$ , a closed cone embedding $a : C_{Z^{'}} T^{'} ↪ N^{'} = N ∣_{Z^{'}}$ , and an operation $c (Z, T, N) : = (p^{' *})^{- 1} a_{*} σ_{Z^{'} / T^{'}} : A_{m} (T^{'}) \to A_{m - d} (Z^{'})$ . This family is bivariant and compatible with arbitrary base restriction. If $N_{0}^{'} \subset N^{'}$ is a vector subbundle through which the cone embedding factors, the operation defined with $N^{'}$ is the operation defined with $N_{0}^{'}$ followed by the top Chern operator of $N^{'} / N_{0}^{'}$ : $c (Z^{'}, T^{'}, N^{'}) = c_{d - d_{0}} (N^{'} / N_{0}^{'}) \cap c (Z^{'}, T^{'}, N_{0}^{'})$ , where $d_{0} = rank N_{0}^{'}$ .

Facts & Assumptions

Given: the Axiom of Choice; a closed immersion $Z ↪ T$ with ideal $I$ and a rank- $d$ bundle $N$ on $Z$ with a surjection $N^{\lor} ↠ I / I^{2}$ ; for every base change $T^{'} \to T$ the closed subscheme $Z^{'} = Z \times_{T} T^{'}$ , its normal cone $C_{Z^{'}} T^{'}$ , the pullback $N^{'} = N ∣_{Z^{'}}$ and the induced surjection $N^{' \lor} ↠ I O_{T^{'}} / (I O_{T^{'}})^{2}$ .

[L1]

The deformation to the normal cone gives the open flat deformation family $D_{i} \to P^{1}$ with special fibre the normal cone and ordinary fibres the ambient scheme and a well-defined specialization operation on Chow groups, natural in the base (Deformation to the normal cone and specialization).

[L2]

Bivariant operations and their axioms; flat pullback along a vector bundle is bijective with inverse $s_{N}^{!}$ ; the zero-section and excess identities hold after every base change (Bivariant Chow operations and bivariant classes, Homotopy invariance for vector bundles, Zero-section Gysin and excess intersection for vector subbundles).

[L3]

Localization and homotopy invariance for Chow groups, and the compatibility of Cartier Gysin with proper pushforward and flat pullback (Localization sequence for Chow groups and homotopy invariance of affine space, Intersection with an invertible sheaf and the first Chern class).

Proof

technique · direct; construct the cone embedding by the Rees-algebra surjection, verify the bivariant axioms for the specialization, and compare base changes through the induced proper morphism of blowups

1.1L1givenalgebra

The cone embedding. Under a base change $T^{'} \to T$ the ideal of $Z^{'}$ is $I O_{T^{'}}$ and its powers are generated by the images of the powers of $I$ , so the Rees algebra of $I O_{T^{'}}$ is a quotient of the base change of the Rees algebra of $I$ ; taking the symmetric algebra of the pulled-back conormal surjection and composing with the canonical map to the associated graded gives $Sym (N^{' \lor}) ↠ {gr}_{I O_{T^{'}}} O_{T^{'}}$ , and hence a closed immersion $a : C_{Z^{'}} T^{'} ↪ N^{'}$ of the normal cone over $Z^{'}$ into the vector bundle. This holds for arbitrary, including nonflat, base change, since only surjectivity of the graded maps is used.

2.1L1L2L3step 1.1algebra

Bivariant axioms. For each base $T^{'}$ the operation is $c (Z, T, N)_{T^{'}} = s_{N^{'}}^{!} a_{*} σ_{Z^{'} / T^{'}}$ , a composite of the well-defined specialization $σ_{Z^{'} / T^{'}} : A_{m} (T^{'}) \to A_{m} (C_{Z^{'}} T^{'})$ of [L1], the proper pushforward $a_{*}$ , and the bijection $s_{N^{'}}^{!} = (p^{' *})^{- 1}$ of [L2]. Proper compatibility: push a lift of the pulled-back class forward along the given proper morphism and use that Cartier Gysin commutes with proper pushforward [L3]. Flat compatibility: pull the lift back and use flat/Cartier compatibility. Divisor compatibility: apply the Cartier Gysin to the lift and use that two Cartier Gysins commute [L3], noting that the ordinary restriction of the result is the lift of the desired input. Base restriction is simply the reindexing of the family. These verifications are exactly the three bivariant axioms of [L2].

3.1L1L2step 1.1step 2.1algebra

Base change. Under a base change the blowup of $I O_{T^{'}}$ admits a closed immersion into the base change of the original blowup, induced by the surjection of Rees algebras of step 1.1, and this immersion is an isomorphism over the ordinary part $A_{T^{'}}^{1}$ ; the induced proper morphism between the two deformation families is therefore an isomorphism over the ordinary fibres, and pushing a lift along it exhibits the two infinity operations as equal after pushforward. Consequently the cone portions of the two operations define the same class after pushforward to $N^{'}$ , and since proper pushforward, open restriction and $(p^{' *})^{- 1}$ are bivariant, the operation is compatible with arbitrary base change.

4.1L2step 2.1algebra∎

The excess factor. Suppose the cone embedding factors through a vector subbundle $N_{0}^{'} \subseteq N^{'}$ of rank $d_{0}$ ; write the cone specialization pushed into $N_{0}^{'}$ as $p_{0}^{' *} β$ for a class $β$ on $Z^{'}$ , using the bundle-pullback bijection for $N_{0}^{'}$ from [L2]. Applying the excess identity $s_{N^{'}}^{!} a_{*}^{'} p_{0}^{' *} β = c_{rank (N^{'} / N_{0}^{'})} (N^{'} / N_{0}^{'}) \cap β$ of [L2] to the vector subbundle inclusion $a^{'} : N_{0}^{'} ↪ N^{'}$ gives that the operation defined with $N^{'}$ equals the operation defined with $N_{0}^{'}$ multiplied by the top Chern operator of the quotient bundle $N^{'} / N_{0}^{'}$ ; this is the asserted excess factor.

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

Refined Gysin operations commute and compose

Statement

All schemes and base changes below are locally of finite type over a fixed field $k$ , and all morphisms are $k$ -morphisms.

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth normal-sequence suppliers. The operations $c (Z, T, N)$ of Gysin specialization is bivariant and compatible with base change commute with every bivariant operation. Given $Z \subset Y \subset T$ and virtual normal bundles fitting into $0 \to N_{Z / Y} \to N_{Z / T} \to N_{Y / T} ∣_{Z} \to 0$ compatible with the conormal maps, $c (Z, T, N_{Z / T}) = c (Z, Y, N_{Z / Y}) c (Y, T, N_{Y / T})$ . For regular embeddings this yields $(j i)^{!} = i^{!} j^{!}$ , after all base changes. If a regular embedding is followed by a smooth projection, its Gysin likewise composes with the projection pullback; in particular a regular section of a smooth morphism gives $s^{!} p^{*} = 1$ .

Facts & Assumptions

Given: the Axiom of Choice; closed immersions $Z \subseteq Y \subseteq T$ with virtual normal bundles $N_{Z / Y}, N_{Y / T}, N_{Z / T}$ in a compatible exact sequence; a regular embedding $s : Z ↪ P$ followed by a smooth projection $P \to T$ with $Z \to T$ flat of the expected relative dimension.

[L1]

The operations $c (Z, T, N)$ of Gysin specialization is bivariant and compatible with base change are bivariant, are compatible with arbitrary base change, and admit the excess formula: if the cone factors through a subbundle, the operation is the corresponding operation multiplied by the top Chern operator of the quotient.

[L2]

Chern classes are operational and central among bivariant operations; the zero-section and excess identities hold (Operational Chern classes and the Whitney formula, Zero-section Gysin and excess intersection for vector subbundles).

[L3]

Cartier Gysin commutes with proper pushforward and flat pullback, and the bivariant equality criterion holds: equality can be tested on fundamental classes of integral schemes over the base, allowing a proper birational modification (Intersection with an invertible sheaf and the first Chern class, Bivariant Chow operations and bivariant classes).

[L4]

Smooth local structure of regular embeddings: the conormal sequence, étale-local coordinate model and regular sequences are as in Smooth immersions, their conormal sequence, deformation charts and smooth sections; blowups and strict transforms are as in Blowup of a scheme along an ideal sheaf.

[L5]

For a nonzero ideal on an integral scheme, the blowup is integral and birational (Blowing up a nonzero ideal on an integral scheme is birational), is proper (Blowups of finite type ideals are locally H-projective, and proper), and its inverse-image ideal is invertible with Cartier zero scheme (The pulled-back center ideal is the relative twist; the exceptional divisor is Cartier). If a product of two finitely generated ideals in a local domain is generated by a nonzero element $c$ , some generator product equals $c$ times a unit; cancellation then shows each factor ideal is principal and generated by a nonzerodivisor. This verifies the simultaneous Cartier reduction used below.

Proof

technique · direct; test on integral bases via the bivariant criterion, reduce to the Cartier case by blowing up, and prove the key cone identity by the deformation charts

1.1L1L2L3L5algebra

Commutation with bivariant operations. Test equality of two bivariant operations on the fundamental class of an integral $T^{'} \to T$ , allowing a proper birational modification by [L3]. If the inverse image of $Z$ in $T^{'}$ is all of $T^{'}$ , the cone is the zero section and $c (Z, T, N)$ is the top Chern operator of the virtual normal bundle by [L1] with $N_{0}^{'} = 0$ , hence central by [L2]. Otherwise blow up the nonzero ideal defining the inverse image of $Z$ : on the blowup the inverse image is an effective Cartier divisor $D$ , the pulled-back conormal surjection maps onto $I_{D} / I_{D}^{2}$ , an invertible sheaf. Locally this surjection onto a free rank-one module splits; dualizing gives a subbundle injection $N_{D / {\tilde{T}}^{'}} ↪ N ∣_{D}$ with locally free quotient $Q$ , and the excess formula of [L1] gives $c = c_{top} (Q) \cap D^{!}$ . The factor $c_{top} (Q)$ commutes with every bivariant operation by [L2] and the Cartier factor $D^{!}$ commutes by the Cartier axiom [L3]; pushing the resulting identity forward along the blowup proves commutation on the original integral base.

1.2L1L2L4algebra

Sections of smooth morphisms. Let $s : Z ↪ P$ be a regular immersion followed by a smooth projection $p : P \to T$ with $p s$ flat of the expected relative dimension. Locally write $T = Spec A$ , $P = Spec B$ , $Z = Spec C$ ; the Koszul resolution of $C$ over $B$ has $A$ -flat terms because $B$ is smooth hence flat over $A$ , and its quotient $C$ is $A$ -flat by hypothesis, so the successive kernels are $A$ -flat by the Tor vanishing criterion; tensoring with any $A^{'}$ therefore stays exact, so the regular equations remain regular after every base change, and the normal cone of the base-changed embedding is the full base-changed normal bundle with fundamental cycle the bundle pullback. Inverting bundle pullback gives $s^{!} p^{*} = (p s)^{*}$ ; when $p s = id$ this is $s^{!} p^{*} = 1$ .

2.1L1L3L4L5step 1.1algebra

Composition. Apply the equality criterion of [L3] and blow up the product of the two ideals defining the inverse images of $Z$ and $Y$ to make both Cartier, splitting off the excess bundles with [L1] so that the outer normal bundle is a line bundle $N_{Y / T}$ over the base; if an inverse image is the whole base the identity reduces to the excess formula of [L1]. Write $o : Y \to C_{Y} T$ for the zero section; the essential cone identity is $o^{!} [C_{Z} T] = [C_{Z} Y]$ as a class in $A_{n - 1} (C_{Z} T \times_{C_{Y} T} Y)$ for integral $T$ of dimension $n$ . On an affine open write $Z = (a)$ and $Y = (b)$ with $b = a c$ and $a, b \neq 0$ ; the $t$ -charts of the two deformations are $R_{a} = A [t, u] / (t u - a)$ and $R_{b} = A [t, v] / (t v - b)$ , both embedded in $A [t, t^{- 1}]$ because their principal ideals are generated by nonzerodivisors, and the homomorphism $R_{b} \to R_{a}$ , $v \mapsto c u$ , defines a morphism from the $t$ -chart of the $Z$ -deformation to that of the $Y$ -deformation which restricts on $t = 0$ to the induced map of normal cones. The strict transform of $Y \times P^{1}$ in the $Z$ -blowup has coordinate ring $R_{a} / ((b) R_{a} : t^{\infty}) = (A / (b)) [t, \overset{ˉ}{a} / t]$ , the contraction of $(b)$ after inverting $t$ (saturation is essential: for $b = a^{2}$ the quotient has relation $u^{2} = 0$ at infinity, not merely $\overset{ˉ}{c} u = 0$ ), and its special fibre is ${gr}_{(\overset{ˉ}{a})} (A / (b)) = C_{Z} Y$ with all nonreduced multiplicities. The strict transform $P_{Y}^{1}$ in the $Y$ -blowup is the Cartier divisor $v = 0$ with infinity restriction the zero section $o$ ; pulling its Cartier operator back to the $Z$ -chart cuts by $c u$ , a nonzero divisor because $R_{a}$ is a domain, and the ordinary parts agree, so the two Cartier classes differ by classes at infinity, which the infinity Cartier Gysin kills because the normal line is trivial ( $j^{!} j_{*} η = c_{1} (1) η = 0$ ). Hence $o^{!} [C_{Z} T] = [C_{Z} Y]$ ; write $p : N_{Z / T} \to Z$ , $p_{1} : N_{Z / Y} \to Z$ , and $e : N_{Z / Y} ↪ N_{Z / T}$ . After the excess reduction, $e$ is the inverse image of the zero section of the outer normal line. Proper/Cartier compatibility pushes the cone identity to $e^{!} a_{*} [C_{Z} T] = a_{1 *} [C_{Z} Y]$ , while the relative-line coordinate calculation gives $e^{!} p^{*} = p_{1}^{*}$ . The classes defining the two operations are characterized by $p^{*} γ = a_{*} [C_{Z} T]$ and $p_{1}^{*} γ_{1} = a_{1 *} [C_{Z} Y]$ ; hence $p_{1}^{*} γ = p_{1}^{*} γ_{1}$ , and the injectivity of $p_{1}^{*}$ proves the required equality, and pushing down the modifications gives the composition formula for regular embeddings.

3.1L1step 2.1step 1.2algebra∎

Graph composition. For morphisms $f : X \to Y$ and $g : Y \to W$ between smooth schemes, work in $X \times Y \times W$ : the two graph pullbacks intersect in the graph of $(f, g f)$ , their equations form regular sequences with independent coordinate directions, and the regular-immersion composition of step 2.1 identifies the iterated operation with the small graph Gysin followed by pullback from $W$ . Comparing with the graph of $g f$ in $X \times W$ uses the regular section $X \times W \to X \times Y \times W$ , $(x, w) \mapsto (x, f (x), w)$ , of a smooth projection, whose Gysin cancels the projection pullback by step 1.2; composition then reduces the small graph operation to the graph of $g f$ . All normal-bundle equalities used are exact sequences, not global splittings.

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Refined Gysin pullback for regular embeddings

Definition

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth normal-sequence suppliers. Fix a field $k$ ; all schemes and base changes below are locally of finite type over $k$ , and all morphisms are $k$ -morphisms. Let $i : X ↪ Y$ be a regular closed embedding of codimension $d$ with normal bundle $N$ . In particular a closed embedding of smooth schemes of constant codimension is regular. For any $f : Y^{'} \to Y$ , put $X^{'} = X \times_{Y} Y^{'}$ and $N^{'} = N ∣_{X^{'}}$ . The pulled-back ideal gives a closed immersion $a : C_{X^{'}} Y^{'} ↪ N^{'}$ . Define $i_{Y^{'}}^{!} = (p^{' *})^{- 1} a_{*} σ_{X^{'} / Y^{'}} : A_{m} (Y^{'}) ⟶ A_{m - d} (X^{'}),$ using Deformation to the normal cone and specialization and Homotopy invariance for vector bundles. No fibre product of deformation spaces over $P^{1}$ with a fictitious map to $Y$ is used. The same construction for a regular locally closed embedding uses restriction to an open in which it is closed; the operations agree under further restriction and extension of cycles, so this is intrinsic.

For $h : Y^{''} \to Y^{'}$ and $h_{X} : X^{''} = X \times_{Y} Y^{''} \to X^{'}$ , properness of $h$ gives $i_{Y^{'}}^{!} h_{*} = (h_{X})_{*} i_{Y^{''}}^{!}$ , and flatness of $h$ of fixed pure relative dimension gives $i_{Y^{''}}^{!} h^{*} = h_{X}^{*} i_{Y^{'}}^{!}$ . These do not require the base-changed embedding to be regular. If $X^{'} ↪ Y^{'}$ is regular with normal bundle $N_{0} \subset N^{'}$ and excess bundle $Q = N^{'} / N_{0}$ , then $i_{Y^{'}}^{!} = c_{d - rank N_{0}} (Q) \cap (i^{'})^{!}$ . In particular $i^{!} i_{*} α = c_{d} (N) \cap α$ . Here $c_{j}$ means the already defined operational Chern operator of Operational Chern classes and the Whitney formula; it later becomes the Chow-ring Chern class on a smooth scheme. Codimension-one Gysin equals Cartier divisor Gysin; refined Gysins commute, and $(j i)^{!} = i^{!} j^{!}$ for composed regular embeddings. They commute with Chern cap operations. For any operational class $c$ on $Y$ and $β \in A_{*} (X)$ , its projection formula is $i_{*} (c ∣_{X} \cap β) = c \cap i_{*} β$ . Under the later smooth operational-ring identification this is the ring formula $i_{*} (i^{*} α \cdot β) = α \cdot i_{*} β$ .

Well-definedness. The construction and the arbitrary-base-change bivariant axioms are proved in Gysin specialization is bivariant and compatible with base change: the specialization $σ_{X^{'} / Y^{'}}$ is well defined on Chow groups by the triviality of the normal line at infinity, the cone embedding exists by the Rees-algebra surjection, and $a_{*}$ and $(p^{' *})^{- 1}$ are the proper pushforward and inverse flat pullback of Homotopy invariance for vector bundles. The excess formula is the corresponding statement of Gysin specialization is bivariant and compatible with base change; for the self-intersection formula apply it to the base change $Y^{'} = X$ , where the ideal is zero, the cone is the zero section and the quotient bundle is $N$ , and use proper compatibility to identify the restricted operation with $i^{!} i_{*}$ . For the agreement with Cartier divisor Gysin in codimension one, test on an integral base cycle: if the cycle is not contained in the divisor its cone is the normal line and the operation is its Cartier fundamental cycle, while if it is contained the zero-cone computation gives $c_{1}$ of the restricted normal line, which are exactly the two cases of the Cartier Gysin of Intersection with an invertible sheaf and the first Chern class. Commutation and composition, including the necessary cone computation with the saturated strict transforms in the deformation charts, are Refined Gysin operations commute and compose. Locality for locally closed embeddings follows because on each cycle the identical ideal and normal bundle give identical operators, and in overlaps the two restrictions agree. The operational projection formula is the proper axiom of a bivariant class (Refined Gysin operations commute and compose); its ring interpretation is provided by the later smooth-ring theorem and is not a prerequisite for this construction.

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

The intersection product and Chow ring of a smooth scheme

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth-immersion and homological suppliers. For a smooth equidimensional finite type $k$ -scheme $X$ of dimension $n$ , $A^{p} (X) = A_{n - p} (X)$ is a commutative graded ring with unit $[X]$ and product $α β = Δ_{X}^{!} (α \times β)$ . The diagonal is a regular immersion (closed if $X$ is separated); use the locally closed extension of Refined Gysin pullback for regular embeddings otherwise. Exterior product on integral cycles is the fundamental cycle $[V \times_{k} W]$ , with generic lengths and all components; over a non-algebraically-closed field this product scheme need not be integral. Exterior product descends to rational equivalence and is associative and symmetric. Every morphism $f : X^{'} \to X$ of smooth finite type $k$ -schemes has a codimension-preserving ring pullback $f^{*} : = Γ_{f}^{!} {pr}_{X}^{*}$ ; it agrees with flat pullback when $f$ is flat and obeys composition. If $X^{'}$ has open equidimensional components $X_{j}^{'}$ of dimensions $n_{j}$ , set $A^{p} (X^{'}) : = ⨁_{j} A_{n_{j} - p} (X_{j}^{'})$ , with componentwise product. For proper $f$ , the restriction $f_{j} : X_{j}^{'} \to X$ has $f_{j, *} : A^{p} (X_{j}^{'}) \to A^{p + n - n_{j}} (X)$ , and $f_{*} (f^{*} α β) = α f_{*} β$ in the total Chow group. Thus the single shift $n - \dim X^{'}$ applies when $X^{'}$ is equidimensional. Operational $c_{j} (E)$ correspond to classes $c_{j} (E) \cap [X]$ and their cap action is multiplication by these classes; in particular $c_{1} (L) α = c_{1} (L) \cap α$ . More generally the operational action on $A_{*} (X)$ agrees with the product after the dimension identification.

Facts & Assumptions

Given: the Axiom of Choice; a smooth equidimensional finite type $k$ -scheme $X$ of dimension $n$ ; its diagonal $Δ_{X} : X \to X \times_{k} X$ and exterior product on cycles.

[L1]

Refined Gysin pullback for regular embeddings is defined, is bivariant, commutes with every bivariant operation, composes, satisfies the excess and self-intersection formulas, and for a regular section of a smooth morphism gives $s^{!} p^{*} = 1$ (Refined Gysin pullback for regular embeddings, Refined Gysin operations commute and compose).

[L2]

The diagonal of a smooth $k$ -scheme is a regular immersion of codimension $n$ ; if $X$ is separated it is a closed immersion, and otherwise a locally closed one; both projections $X \times_{k} X \to X$ are smooth (Smooth morphism of schemes, Fibre product of schemes, Refined Gysin operations commute and compose).

[L3]

Cycles, fundamental cycles, flat pullback and proper pushforward are as in Cycles of coherent sheaves and of closed subschemes, with flat pullback, Flat pullback of cycles and of rational equivalence and Proper pushforward of cycles and the norm formula; exterior product on cycles is the assignment on integral cycles $[V] \times [W] \mapsto [V \times_{k} W]$ .

[L4]

Operational Chern classes and their cap action are defined on singular schemes and satisfy the Whitney and section formulas (Operational Chern classes and the Whitney formula).

Proof

technique · direct; define the exterior product by flat pullback and proper pushforward, define the product by the diagonal Gysin, and verify the ring axioms by the composition and commutation theorems for refined Gysin

1.1L1L3givenalgebra

Exterior product. For integral closed subschemes $V \subseteq X$ , $W \subseteq X^{'}$ define $[V] \times [W] : = [V \times_{k} W]$ , the fundamental cycle of the product, taken with all irreducible components and their generic lengths by [L3]; this is the product of the cycle classes and is bilinear. For a fixed integral $W \subseteq X^{'}$ , it is the flat pullback along $X \times W \to X$ followed by the closed-immersion pushforward $X \times W ↪ X \times X^{'}$ ; the symmetric description handles the other variable. Hence descends to rational equivalence in either variable and commutes with all bivariant operations by [L1]; symmetry of the construction and of the generic lengths makes the product symmetric, and associativity is checked on fundamental cycles of triple products, where both iterated flat pullbacks compute the same generic length by associativity of tensor products and the flat length multiplicity computation. Extension is bilinear and the components are retained with their multiplicities, so no integrality hypothesis on the field is used.

2.1L1L2step 1.1algebra

The product. The diagonal $Δ_{X}$ is a regular immersion of codimension $n$ by [L2], so the refined Gysin $Δ_{X}^{!}$ is defined and graded; set $α β : = Δ_{X}^{!} (α \times β)$ . Associativity follows by comparing the two codimension- $n$ diagonals $Δ_{12}, Δ_{23} \subseteq X^{3}$ : the iterated Gysins both equal the small diagonal Gysin by the composition theorem of [L1], and exterior-product compatibility moves each inner Gysin into $X^{3}$ ; commutativity follows from the symmetry of $Δ_{X}$ and of the exterior product in step 1.1. The unit is $[X]$ : $α \times [X] = {pr}_{1}^{*} α$ for the smooth first projection, and $Δ^{!} {pr}_{1}^{*} = 1$ by the smooth-section identity of [L1] (the diagonal is a section of the smooth projection ${pr}_{1}$ ), so $Δ^{!} ({pr}_{1}^{*} α) = α$ ; the other unit is symmetric.

3.1L1L2step 2.1algebra

Ring pullback. For a morphism $f : X^{'} \to X$ of smooth schemes define $f^{*} : = Γ_{f}^{!} {pr}_{X}^{*}$ , where $Γ_{f} \subseteq X^{'} \times_{k} X$ is the graph, a regular immersion because it is a section of the smooth projection $X^{'} \times_{k} X \to X^{'}$ by [L2], and ${pr}_{X}^{*}$ is flat pullback. When $f$ is flat this agrees with the flat pullback: the graph Gysin commutes with flat pullback and $f^{*}$ is characterized on test classes by the same computation, and composition of ring pullbacks holds by the composition theorem for refined Gysin applied to the graphs and the projection identity of [L1]. Since the graph is a section of a smooth morphism, $Γ_{f}^{!} {pr}_{X^{'}}^{*} = 1$ , which identifies $f^{*}$ with the codimension-preserving pullback of the smooth-ring statement.

4.1L1step 2.1step 3.1algebra

Projection formula. For proper $f : X^{'} \to X$ and classes $α, β$ , write $α$ as the operational class $α \cap - = α \times -$ composed with the diagonal, i.e. $α = c [X]$ for the bivariant class $c$ given by exterior product with $α$ and diagonal Gysin; then $f_{*} (f^{*} α β) = f_{*} (c β) = c f_{*} β = α f_{*} β$ by the proper axiom of bivariant classes and the identification of step 2.1. This is the displayed projection formula; on each equidimensional component $X_{j}^{'}$ the dimension grading gives the shift $n - n_{j}$ . A smooth finite type scheme has finitely many open equidimensional components, since its regular local rings make its irreducible components disjoint; all graph and operational computations apply componentwise.

5.1L1L4step 2.1step 4.1algebra∎

Operational Chern classes. For $α \in A_{*} (X)$ define an operation on a test morphism $h : T \to X$ by $c_{α} (β) = Γ_{h}^{!} (α \times β)$ , with the graph in $X \times T$ . The graph is a regular section of the smooth projection to $T$ , even for singular $T$ , and [L1] and step 1.1 give the proper, flat and Cartier axioms. Its value on $[X]$ is $α$ by the unit computation. Conversely, for an operational class $c$ and integral $V \subseteq T$ , use the flat projection $X \times V \to X$ and the closed immersion $X \times V \to X \times T$ to get $c ([X] \times [V]) = (c [X]) \times [V]$ . Commutation of $c$ with graph Gysin by [L1], and the smooth-section identity $Γ_{h}^{!} ([X] \times [V]) = [V]$ , give $c [V] = Γ_{h}^{!} ((c [X]) \times [V])$ . Extend linearly to every class. Evaluation on $[X]$ and $α \mapsto c_{α}$ are therefore inverse, and on $T = X$ the action of $c$ is multiplication by $c [X]$ . For the classes $c_{j} (E)$ of [L4] this is multiplication by $c_{j} (E) \cap [X]$ , compatible with Whitney and section formulas.

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

Naturality of the Chow ring and the projection formula

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth-immersion and homological suppliers. For every morphism $f : X \to Y$ of smooth equidimensional finite type schemes over $k$ , $f^{*} : A^{*} (Y) \to A^{*} (X)$ is the graph Gysin pullback, preserves codimension and the unit, is a ring homomorphism, and obeys $(g f)^{*} = f^{*} g^{*}$ . If $f$ is flat it is precisely the previously defined flat pullback. For proper $f$ , proper pushforward shifts codimension by $\dim Y - \dim X$ , obeys $(g f)_{*} = g_{*} f_{*}$ for proper composites, and $f_{*} (f^{*} α \cdot β) = α \cdot f_{*} β$ . Flat/proper composition laws are asserted under their respective hypotheses, not by identifying the two kinds of operation.

Facts & Assumptions

Given: the Axiom of Choice; smooth equidimensional finite type $k$ -schemes $X, Y$ and a morphism $f : X \to Y$ ; for the last assertions a composable morphism $g : Y \to Z$ .

[L1]

The Chow ring structure: $A^{p} (X) = A_{\dim X - p} (X)$ is a commutative graded ring with product $Δ_{X}^{!} (α \times β)$ , unit $[X]$ , graph pullback $f^{*} = Γ_{f}^{!} {pr}_{Y}^{*}$ , and for proper $f$ the projection formula $f_{*} (f^{*} α β) = α f_{*} β$ (The intersection product and Chow ring of a smooth scheme).

[L2]

Graph Gysin pullback is a regular-section Gysin, hence commutes with flat pullback and composes; the graph is a regular immersion and a section of the smooth projection to the source. When $f$ is flat, the regular-immersion-followed-by-smooth-projection identity applied to $Γ_{f} : X ↪ X \times_{k} Y$ and ${pr}_{Y}$ gives $Γ_{f}^{!} {pr}_{Y}^{*} = f^{*}$ for the flat pullback (Refined Gysin operations commute and compose).

[L3]

Proper pushforward of cycles is functorial under composition and shifts degrees by the dimension difference; flat pullback is functorial and preserves codimension (Proper pushforward of cycles and the norm formula, Flat pullback of cycles and of rational equivalence).

Proof

technique · direct; translate each statement into the operational description of the pullback and apply the composition and commutation theorems for refined Gysin

1.1L1L2givenalgebra

Pullback is a unital ring homomorphism. By [L1] the graph pullback $f^{*} = Γ_{f}^{!} {pr}_{Y}^{*}$ is a composite of refined Gysin operations, which preserve codimension by [L2]; it is unital because $Γ_{f}^{!} {pr}_{Y}^{*} [Y] = [Γ_{f}] = [X]$ up to the canonical identification of the graph with $X$ , and the smooth-section identity gives $Γ_{f}^{!} {pr}_{X}^{*} = 1$ . To see that $f^{*}$ is multiplicative, write $α = c [Y]$ and $β = d [Y]$ in the operational description of [L1]: restriction of operational classes is a ring map by [L2], so $f^{*} (α β) = f^{*} (c d [Y]) = c d [X] = f^{*} α f^{*} β$ . Composition $(g f)^{*} = f^{*} g^{*}$ follows from the composition theorem for refined Gysin applied to the composable graphs, with the projection identity of [L2].

2.1L2L3step 1.1algebra

Agreement with flat pullback. If $f$ is flat, apply the regular-immersion/smooth-projection identity of [L2] to $Γ_{f} : X ↪ X \times_{k} Y$ followed by ${pr}_{Y} : X \times_{k} Y \to Y$ . Its composite is $f$ , flat of relative dimension $\dim X - \dim Y$ , so the identity gives $Γ_{f}^{!} {pr}_{Y}^{*} = f^{*}$ for the flat pullback; hence $f^{*}$ agrees with the flat pullback of [L3] on classes of the stated dimensions.

3.1L1L3step 1.1algebra∎

Proper pushforward. If $f$ is proper, [L3] gives functoriality $(g f)_{*} = g_{*} f_{*}$ and the codimension shift $\dim Y - \dim X$ on nonzero cycle classes, since a proper pushforward of a $p$ -dimensional class is supported on images of dimension at most $p$ and the norm-degree formula is multiplicative in towers. The projection formula is the corresponding statement of [L1], with the identification $A^{p} = A_{\dim - p}$ : writing $α = c [Y]$ in operational form, $f_{*} (c β) = c f_{*} β$ is the proper axiom of bivariant classes, which is exactly $f_{*} (f^{*} α β) = α f_{*} β$ .

DefinitionDefinition: Literature-sourcedProof: Not applicableOpen item page →

Chern classes of a vector bundle on a smooth scheme

Definition

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth-immersion and homological suppliers. Let $k$ be a field and let $X$ be a smooth equidimensional scheme of finite type over $k$ of dimension $n$ (Smooth morphism of schemes, Relative dimension of a smooth morphism at a point), with Chow ring $A^{*} (X)$ (The intersection product and Chow ring of a smooth scheme). Let $E$ be a finite locally free $O_{X}$ -module of rank $r \geq 1$ (Locally free sheaves of finite rank), with projective bundle $π : P (E) \to X$ and $ξ = c_{1} (O (1))$ (The projective bundle formula for Chow groups).

Definition. By the projective bundle formula, $ξ^{r} \in A^{r} (P (E))$ has a unique expression $ξ^{r} + \sum_{i = 1}^{r} (- 1)^{i} c_{i} (E) ξ^{r - i} = 0 in A^{*} (P (E)),$ with $c_{i} (E) \in A^{i} (X)$ pulled back from $X$ ; the classes $c_{i} (E)$ are the Chern classes of $E$ . Set $c_{0} (E) : = 1 \in A^{0} (X)$ , $c_{i} (E) : = 0$ for $i > r$ , and call $c (E) : = 1 + c_{1} (E) + \dots + c_{r} (E)$ the total Chern class.

For the zero bundle define $c (0) = 1$ and $c_{i} (0) = 0$ for $i > 0$ , as for the rank-zero operational classes; no projective bundle of rank zero is used.

Basic properties. (i) (Normalization) For an invertible sheaf $L$ : $c (L) = 1 + c_{1} (L)$ with $c_{1} (L) \in A^{1} (X)$ the first Chern class of Intersection with an invertible sheaf and the first Chern class; in particular $c_{1} (O_{X} (D)) = [D]$ for an effective Cartier divisor $D$ . (ii) (Naturality) For a morphism $f : X^{'} \to X$ of smooth equidimensional $k$ -schemes, $c_{i} (f^{*} E) = f^{*} c_{i} (E)$ . (iii) (Vanishing) $c_{i} (E) = 0$ for $i > rank E$ and $c_{i} (E) \in A^{i} (X)$ . (iv) (Direct sums of line bundles) If $E ≅ L_{1} \oplus \dots \oplus L_{r}$ then $c (E) = \prod_{i = 1}^{r} (1 + c_{1} (L_{i}))$ . (v) (Top class) $c_{r} (E) \cap [X] = [Z (s)]$ for a regular section $s$ of $E$ with $\dim Z (s) = n - r$ ; in particular $c_{r} (E)$ is the class of the zero locus of a regular section.

Well-definedness. The classes are defined by applying the operational Chern operators of Operational Chern classes and the Whitney formula to the fundamental class $[X]$ and using the operational-ring isomorphism of The intersection product and Chow ring of a smooth scheme, which identifies the operational action with multiplication: their defining projective bundle relation is exactly the displayed relation, and the basis part of The projective bundle formula for Chow groups gives existence and uniqueness of the coefficients. Normalization, naturality under arbitrary morphisms of smooth schemes, rank vanishing, the Whitney and line-summand formulas and the regular-section formula (including its generic multiplicities and the case of a non-reduced zero scheme) are already established for the operational classes; restriction of operational classes is the graph pullback under the operational isomorphism, by Naturality of the Chow ring and the projection formula, so they become the claimed ring identities. Only the injectivity of projective-bundle pullback is used, not injectivity of arbitrary $f^{*}$ .

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

Additivity, naturality and the splitting principle for Chern classes

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth-immersion and homological suppliers. Let $k$ be a field, $X$ a smooth equidimensional $k$ -scheme of finite type, and let $0 \to E^{'} \to E \to E^{''} \to 0$ be a short exact sequence of finite locally free $O_{X}$ -modules (Locally free sheaves of finite rank). Ranks are allowed to be locally constant: define Chern classes componentwise on the finitely many open-and-closed rank loci, using Chern classes of a vector bundle on a smooth scheme on each constant-rank locus. Then:

(Whitney additivity) $c (E) = c (E^{'}) c (E^{''})$ in the Chow ring $A^{*} (X)$ (Chern classes of a vector bundle on a smooth scheme, The intersection product and Chow ring of a smooth scheme).
(Naturality) If $f : X^{'} \to X$ is a flat morphism of smooth equidimensional $k$ -schemes of finite type, then $c_{i} (f^{*} E) = f^{*} c_{i} (E)$ for all $i$ .
(Splitting principle) There exists a morphism $f : X^{'} \to X$ , a composition of projective bundles over $X$ , with $X^{'}$ smooth and equidimensional, such that $f^{*} : A^{*} (X) \to A^{*} (X^{'})$ is injective. On each rank locus $X_{r} = {x : {rank}_{x} E = r}$ , its inverse image admits a filtration $0 = E_{0} \subseteq E_{1} \subseteq \dots \subseteq E_{r} = f^{*} E$ with successive quotients invertible sheaves $L_{1}, \dots, L_{r}$ ; consequently $f^{*} c (E) = \prod_{i = 1}^{r} (1 + c_{1} (L_{i}))$ . Any polynomial identity among Chern classes that is proved on every rank locus after replacing $E$ by such a filtered bundle and $c_{i} (E)$ by the elementary symmetric functions $e_{i} (c_{1} (L_{j}))$ holds for $E$ itself.
(Consequences) $c_{i} (E^{\lor}) = (- 1)^{i} c_{i} (E)$ ; $c_{1} (L \otimes M) = c_{1} (L) + c_{1} (M)$ for invertible sheaves; $c_{i} (E) = 0$ for $i > r$ on $X_{r}$ ; and for an exact sequence of vector bundles the total Chern classes multiply (this is (1)).

Facts & Assumptions

Given: the Axiom of Choice; a smooth equidimensional finite type $k$ -scheme $X$ ; an exact sequence $0 \to E^{'} \to E \to E^{''} \to 0$ of finite locally free $O_{X}$ -modules.

[L1]

The operational Chern operators satisfy the Whitney formula, normalization, compatibility with base restriction and the splitting principle by iterated projective bundles with injective flat pullback, and the regular-section formula (Operational Chern classes and the Whitney formula, Chern classes of a vector bundle on a smooth scheme).

[L2]

The operational action on a smooth scheme agrees with multiplication in the Chow ring, so operational identities become ring identities; ring pullback is the graph Gysin pullback and agrees with flat pullback for flat morphisms (The intersection product and Chow ring of a smooth scheme, Naturality of the Chow ring and the projection formula).

[L3]

Projective bundle pullback is injective and flat pullback on Chow groups is functorial (The projective bundle formula for Chow groups, Flat pullback of cycles and of rational equivalence, Refined Gysin pullback for regular embeddings).

[L4]

First Chern classes of invertible sheaves are computed by rational sections and their divisors; the dual of a line bundle has the negative first Chern class (Intersection with an invertible sheaf and the first Chern class).

Proof

technique · direct; transport the operational Whitney and splitting arguments to the Chow ring through the operational isomorphism

1.1L1L2givenalgebra

Whitney additivity. By [L1] the operational classes of $E$ satisfy $c (E) = c (E^{'}) c (E^{''})$ : after simultaneous flag pullback the two filtrations concatenate to a line filtration of the middle bundle, whose Chern product is determined by successive Cartier hyperplanes; injectivity of projective-bundle pullback descends the identity; under the operational isomorphism of [L2] this identity of operators is the ring identity $c (E) = c (E^{'}) c (E^{''})$ in $A^{*} (X)$ .

1.2L1L2givenalgebra

Naturality. For a flat morphism $f : X^{'} \to X$ of smooth equidimensional finite type $k$ -schemes, the pullback $f^{*}$ on the Chow ring agrees with flat pullback by [L2], and flat pullback commutes with the operational cap actions by [L1]; applying this to the definitions $c_{i} (E) = c_{i} (E) \cap [X]$ and using the projection formula of [L2] gives $f^{*} c_{i} (E) = c_{i} (f^{*} E)$ .

2.1L1L3step 1.1algebra

The splitting principle. Since $X$ is quasi-compact, the locally constant rank of $E$ has finite image; its rank loci $X_{r}$ are open and closed. Chern classes, cycle groups and rational equivalence decompose over this finite disjoint union, so [L1] applies on every constant-rank locus (and on a common refinement of the three rank decompositions for Whitney additivity). If $X$ is empty, its Chow ring is zero and take the identity with empty filtrations. Otherwise let $R$ be the largest occurring rank. If $R \leq 1$ , take $f = id$ , with the empty filtration on $X_{0}$ and the one-step filtration on $X_{1}$ . Otherwise build a tower indexed by $j = R, R - 1, \dots, 2$ . Over a locus of original rank $r \geq j$ , projectivize the current kernel bundle of rank $j$ and replace it by the kernel of its tautological line quotient. Over a locus with $r < j$ , projectivize the trivial bundle of rank $j$ and leave the pulled-back $E$ unchanged. At each stage these bundles glue across the open-and-closed loci to a bundle of constant rank $j$ , so every projection is smooth of constant relative dimension $j - 1$ and has injective flat pullback by [L3]. The resulting $X^{'}$ is smooth equidimensional of dimension $\dim X + R (R - 1) / 2$ . On each inverse image of $X_{r}$ , the successive tautological kernels, reversed in order, yield a filtration with exactly $r$ invertible quotients. Repeated Whitney additivity gives $f^{*} c (E) ∣_{f^{- 1} X_{r}} = \prod_{i = 1}^{r} (1 + c_{1} (L_{i}))$ ; the composite pullback is injective, so polynomial identities verified on every rank locus descend to $X$ .

3.1L1L4step 1.1step 2.1algebra∎

Consequences. The dual of the filtration has successive quotients $L_{i}^{\lor}$ with $c_{1} (L_{i}^{\lor}) = - c_{1} (L_{i})$ by [L4], so multiplying $\prod_{i} (1 - c_{1} (L_{i}))$ gives $c_{i} (E^{\lor}) = (- 1)^{i} c_{i} (E)$ after injectivity; the tensor identity for line bundles follows by multiplying rational sections and adding their Cartier divisors by [L4]; the vanishing $c_{i} (E) = 0$ for $i > rank E$ is the defining relation of [L1], and the multiplicativity for exact sequences is step 1.1.

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

The Chern character and the Todd class

Definition

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. Let $k$ be a field and let $X$ be a smooth equidimensional $k$ -scheme of finite type (Smooth morphism of schemes), with Chow ring $A^{*} (X)$ (The intersection product and Chow ring of a smooth scheme) and vector-bundle group $K^{0} (X)$ (Grothendieck groups of coherent sheaves and of vector bundles on a scheme). When $X$ is additionally quasi-projective, Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes identifies this group with coherent $K_{0} (X)$ ; that extra hypothesis is required for the coherent-group extension. Write $A^{*} (X)_{Q} : = A^{*} (X) \otimes_{Z} Q$ . If $n = \dim X$ , $A^{i} (X) = A_{n - i} (X) = 0$ for $i > n$ , since there are no negative-dimensional integral cycles. Every series below is evaluated only through degree $n$ ; in particular positive-codimension elements are nilpotent. For the rank-zero bundle the empty root list gives $ch (0) = 0$ and $td (0) = 1$ .

Chern roots. The Chern roots of a finite locally free $O_{X}$ -module $E$ of rank $r$ are formal symbols $x_{1}, \dots, x_{r}$ with the property that the elementary symmetric functions in the $x_{i}$ are the Chern classes: $e_{i} (x_{1}, \dots, x_{r}) = c_{i} (E)$ (Chern classes of a vector bundle on a smooth scheme); by the splitting principle every symmetric polynomial expression in the $x_{i}$ with rational coefficients defines a well-defined element of $A^{*} (X)_{Q}$ (Additivity, naturality and the splitting principle for Chern classes).

Chern character. Define $ch (E) : = \sum_{j = 1}^{r} e^{x_{j}} = \sum_{m \geq 0} \frac{1}{m!} p_{m} (c_{1}, \dots, c_{m}) \in A^{*} (X)_{Q},$ where $p_{0} : = r = rank E$ , while for $m \geq 1$ the power sum $p_{m} (c_{1}, \dots, c_{m}) = \sum_{j} x_{j}^{m}$ is a universal polynomial in the Chern classes. Thus ${ch}_{0} (E) = r$ ; explicitly $ch = r + c_{1} + \frac{1}{2} (c_{1}^{2} - 2 c_{2}) + \frac{1}{6} (c_{1}^{3} - 3 c_{1} c_{2} + 3 c_{3}) + \dots$ (the denominators are why we tensor with $Q$ ).

Todd class. Define $td (E) : = \prod_{j = 1}^{r} \frac{x_{j}}{1 - e^{- x_{j}}} \in A^{*} (X)_{Q};$ explicitly $td (E) = 1 + \frac{1}{2} c_{1} + \frac{1}{12} (c_{1}^{2} + c_{2}) + \frac{1}{24} c_{1} c_{2} + \dots$ . For a smooth $X$ the Todd class of $X$ is $td (T_{X})$ for the tangent bundle $T_{X} = Hom (Ω_{X / k}^{1}, O_{X})$ (Sheaf of relative Kähler differentials, Differentials of a smooth morphism).

Line bundles. For an invertible sheaf $L$ with $x = c_{1} (L)$ : $ch (L) = e^{x}$ and $td (L) = x / (1 - e^{- x}) = 1 + \frac{1}{2} x + \frac{1}{12} x^{2} - \dots$ .

Well-definedness. After truncation at degree $n$ , the positive-degree components of the Chern character and all components of the Todd class are symmetric polynomials with rational coefficients in the Chern roots, hence universal polynomial expressions in their elementary symmetric functions, the Chern classes of Chern classes of a vector bundle on a smooth scheme. The degree-zero component of the Chern character is the rank $r$ . Thus the Chern character depends only on the rank and Chern classes of $E$ , whereas the Todd class depends only on its Chern classes. For locally constant ranks these formulas are interpreted componentwise on the finite open-and-closed rank loci. By the splitting principle of Additivity, naturality and the splitting principle for Chern classes, these values are independent of the choice of flag bundle or filtration: any two filtrations have the same rank and elementary symmetric functions, and hence give the same polynomial values. Rational coefficients are needed for the exponential and geometric expansions, which is why the target is $A^{*} (X) \otimes_{Z} Q$ . The tangent bundle is finite locally free of rank $\dim X$ by Differentials of a smooth morphism, so its Chern roots and Todd class are defined; the Todd class is multiplicative in exact sequences, whereas the Chern character is additive, and the identification of $K^{0} (X)$ with $K_{0} (X)$ on a quasi-projective $X$ lets the character be evaluated on coherent classes through the finite locally free resolutions of Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes.

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

Additivity and multiplicativity of the Chern character and Todd class

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. In the setting of The Chern character and the Todd class let $0 \to E^{'} \to E \to E^{''} \to 0$ be a short exact sequence of finite locally free $O_{X}$ -modules (Locally free sheaves of finite rank). Then:

$ch (E) = ch (E^{'}) + ch (E^{''})$ in $A^{*} (X)_{Q}$ ; hence $ch$ is additive in exact sequences and descends to a group homomorphism $ch : K^{0} (X) ⟶ A^{*} (X)_{Q} .$
$td (E) = td (E^{'}) td (E^{''})$ ; hence $td$ is multiplicative in exact sequences and depends only on the class of $E$ in $K^{0} (X)$ .
For finite locally free $E, F$ : $ch (E \otimes F) = ch (E) ch (F)$ , ${ch}_{m} (E^{\lor}) = (- 1)^{m} {ch}_{m} (E)$ ; equivalently $ch (E^{\lor}) = \sum e^{- x_{j}}$ when $ch (E) = \sum e^{x_{j}}$ , and $ch (O_{X}) = rank = 1$ . Hence $ch : K^{0} (X) \to A^{*} (X)_{Q}$ is a ring homomorphism with the vector-bundle tensor product.
For a flat morphism $f : X^{'} \to X$ of smooth equidimensional $k$ -schemes: $f^{*} ch (E) = ch (f^{*} E)$ and $f^{*} td (E) = td (f^{*} E)$ ; and $td (T_{X \times_{k} X^{'}}) = {pr}_{1}^{*} td (T_{X}) \cdot {pr}_{2}^{*} td (T_{X^{'}})$ .
For line bundles: $ch (L_{1} \oplus \dots \oplus L_{r}) = \sum e^{c_{1} (L_{j})}$ and $td (L) = \frac{c_{1} (L)}{1 - e^{- c_{1} (L)}}$ (formal series); the terms of degree $> \dim X$ vanish. If $X$ is quasi-projective, transport these maps along the finite-resolution isomorphism $K^{0} (X) = K_{0} (X)$ . The product on coherent $K_{0}$ is the transported vector-bundle product, equivalently the alternating Tor product; ordinary tensor product of two arbitrary coherent sheaves is not assumed additive in exact sequences. Todd values of virtual classes use inverses, which exist because their degree-zero component is one and positive codimension is nilpotent.

Facts & Assumptions

Given: the Axiom of Choice; a smooth equidimensional finite type $k$ -scheme $X$ ; an exact sequence $0 \to E^{'} \to E \to E^{''} \to 0$ of finite locally free sheaves; further finite locally free sheaves $F$ , $L_{1}, \dots, L_{r}$ ; a flag bundle $f : X^{'} \to X$ as in the splitting principle.

[L1]

The splitting principle: there is a composition of projective bundles $f : X^{'} \to X$ with $f^{*}$ injective on the Chow ring after tensoring with $Q$ , such that on each rank locus $f^{*} E$ has a filtration with invertible quotients; identities proved on the common open-and-closed rank-locus refinement of the finitely many bundles descend by injectivity (Additivity, naturality and the splitting principle for Chern classes, The Chern character and the Todd class).

[L2]

Chern classes of pullbacks are the pullbacks of Chern classes for flat morphisms, and the Chow ring pullback is a ring homomorphism (Naturality of the Chow ring and the projection formula, The intersection product and Chow ring of a smooth scheme, Chern classes of a vector bundle on a smooth scheme).

[L3]

On each constant-rank locus, ${ch}_{0} (E) = rank E$ , while each positive-degree component of $ch (E)$ and each component of $td (E)$ is a universal rational polynomial in the Chern classes, with ${td}_{0} (E) = 1$ . Thus the Chern character is determined by the rank function together with the Chern classes, and the Todd class by the Chern classes. The tangent bundle is finite locally free (The Chern character and the Todd class).

Proof

technique · direct; split the bundles by a flag bundle and read the identities off the roots, then descend by injectivity of the pullback

1.1L1L3givenalgebra

Additivity of the Chern character. Work on the common finite open-and-closed refinement of the rank loci of $E$ , $E^{'}$ , $E^{''}$ and $F$ . Chow rings and the claimed identities decompose over this refinement. On each locus all ranks are constant, including rank zero with an empty root list. By [L1] there is a flag bundle $f : X^{'} \to X$ such that $f^{*} E^{'}$ and $f^{*} E^{''}$ have filtrations with invertible quotients, and the roots of $f^{*} E$ are the union of the roots of $f^{*} E^{'}$ and of $f^{*} E^{''}$ : the successive quotients of a filtration of $f^{*} E$ refine to the union of the two filtrations. The degree-zero equality is $rank E = rank E^{'} + rank E^{''}$ . Since the sum of $e^{x}$ over the union of the two root lists is the sum over the separate lists, $ch (f^{*} E) = ch (f^{*} E^{'}) + ch (f^{*} E^{''})$ ; applying $f^{*}$ to the expressions in rank and Chern classes (pullback preserves rank) and using injectivity of $f^{*}$ on $A^{*} (X)_{Q}$ gives additivity for $E$ . Hence $ch$ respects the exact-sequence relations and descends to the group completion $K^{0} (X) \to A^{*} (X)_{Q}$ .

1.2L1L3givenalgebra

Multiplicativity of the Todd class. On each rank locus with the same flag bundle, the roots of $f^{*} E$ are the union of those of $f^{*} E^{'}$ and $f^{*} E^{''}$ , and the product $\prod_{j} x_{j} / (1 - e^{- x_{j}})$ over the union is the product of the two partial products; injectivity of $f^{*}$ descends the identity $td (E) = td (E^{'}) td (E^{''})$ . Thus $td$ is multiplicative on exact sequences and factors through $K^{0} (X)$ ; its values are units because the degree-zero component is $1$ , so it extends to virtual classes by inversion of the full Todd unit, using the finite geometric series in its nilpotent positive-degree part.

1.3L1L3givenalgebra

Tensor products and duality. On each common rank locus, splitting both $E$ and $F$ by a common flag bundle, the roots of the tensor product are the pairwise sums $x_{i} + y_{j}$ and $e^{x_{i} + y_{j}} = e^{x_{i}} e^{y_{j}}$ , so $ch (E \otimes F) = ch (E) ch (F)$ after descent; the dual has roots $- x_{j}$ , and the degree- $m$ part of $e^{- x}$ is $(- 1)^{m}$ times that of $e^{x}$ , giving ${ch}_{m} (E^{\lor}) = (- 1)^{m} {ch}_{m} (E)$ in the graded sense, with no ungraded assertion; $ch (O_{X}) = 1$ is the single-root case with root $0$ . Consequently $ch$ is a unital ring homomorphism on $K^{0} (X)$ with the vector-bundle tensor product.

1.4L2L3givenalgebra

Naturality and the tangent bundle. Pullback preserves the rank function, since it takes a local trivialization $E ∣_{U} ≅ O_{U}^{r}$ to one of rank $r$ on $f^{- 1} (U)$ . By [L2] it also preserves the Chern classes, and by [L3] the components of the Chern character are universal polynomials in rank and Chern classes, while those of the Todd class are universal polynomials in the Chern classes; hence $f^{*} ch (E) = ch (f^{*} E)$ and $f^{*} td (E) = td (f^{*} E)$ for flat $f$ . For the product $X \times_{k} X^{'}$ the tangent bundle is the direct sum of the two pullbacks of the tangent bundles, so the Todd class multiplies: $td (T_{X \times_{k} X^{'}}) = {pr}_{1}^{*} td (T_{X}) {pr}_{2}^{*} td (T_{X^{'}})$ .

2.1L1L3step 1.3givenalgebra∎

Line bundles and coherent classes. For a direct sum of line bundles the identities are the definitions with $x_{j} = c_{1} (L_{j})$ ; the series truncate because $A^{m} (X) = 0$ above the dimension. If $X$ is quasi-projective, the finite-resolution isomorphism $K^{0} (X) = K_{0} (X)$ transports the additive and multiplicative structure, the product on coherent classes being the alternating Tor product; the ordinary tensor product of two arbitrary coherent sheaves is not assumed to be additive in exact sequences, and Todd values of virtual classes use inverses of the full Todd units.

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

Riemann-Roch for projective-space projections

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. Let $k$ be an algebraically closed field, let $Y$ be a nonsingular equidimensional quasi-projective $k$ -scheme of finite type (Smooth morphism of schemes, Classical and scheme smoothness over a perfect field), let $n \geq 0$ and let $p : P_{k}^{n} \times_{k} Y ⟶ Y$ be the projection. For every $α \in K_{0} (P^{n} \times_{k} Y)$ (Grothendieck groups of coherent sheaves and of vector bundles on a scheme): $ch (p_{!} (α)) \cdot td (T_{Y}) = p_{*} (ch (α) \cdot td (T_{P^{n} \times_{k} Y})) in A^{*} (Y)_{Q},$ where $ch, td$ are the Chern character and Todd class of The Chern character and the Todd class, $p_{!}$ is the K-theory pushforward (Pushforward of coherent sheaves in algebraic K-theory, Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes) and $p_{*}$ is the proper pushforward on Chow groups (Proper pushforward of cycles and the norm formula). Equivalently, writing $τ_{X} (α) = ch (α) td (T_{X})$ , the diagram $\begin{array}{ccc} K_{0} (P^{n} \times_{k} Y) & \overset{τ}{\to} & A^{*} (P^{n} \times_{k} Y)_{Q} \\ ↓ p_{!} & ↓ p_{*} \\ K_{0} (Y) & \overset{τ}{\to} & A^{*} (Y)_{Q} \end{array}$ commutes. The case $Y = Spec k$ is Hirzebruch-Riemann-Roch for $P^{n}$ , and the case of a general projective-bundle projection is obtained by the same argument applied to the projective bundle formula.

Facts & Assumptions

Given: the Axiom of Choice; an algebraically closed field $k$ ; a nonsingular equidimensional quasi-projective finite type $k$ -scheme $Y$ ; the projection $p : P_{k}^{n} \times_{k} Y \to Y$ with $h = c_{1} (O_{P^{n}} (1))$ .

[L1]

$K_{0} (P^{n} \times_{k} Y)$ is generated by $p_{Y}^{*} β \otimes p_{P}^{*} [O (m)]$ , $0 \leq m \leq n$ , and $K^{0} = K_{0}$ on the smooth quasi-projective total space (Relative projective bundles: K-theory generation by tautological twists, Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes, K-theory of projective space and of projective bundles).

[L2]

The Chern character is additive and multiplicative, is natural for flat pullback, and the Todd class is multiplicative in exact sequences (Additivity and multiplicativity of the Chern character and Todd class, The Chern character and the Todd class).

[L3]

Projective pushforward of Chow classes is functorial and satisfies the projection formula (Proper pushforward of cycles and the norm formula, Naturality of the Chow ring and the projection formula, The intersection product and Chow ring of a smooth scheme); degrees on projective space are computed by the degree isomorphism of Chow groups of projective space and flat pullback is compatible with degree (Flat pullback of cycles and of rational equivalence).

Proof

technique · direct; reduce to the generators of $K_0$ by additivity, compute the K pushforward of the twists and the Chow pushforward of the corresponding integrands by the formal residue, and extend to projective bundles by the flag/divided-difference computation

1.1L1L2givenalgebra

K-theory pushforward of the generators. By [L1] it suffices to verify the identity on the generators $p_{Y}^{*} β \otimes p_{P}^{*} [O (m)]$ for $β \in K_{0} (Y)$ and $0 \leq m \leq n$ . The K-theory projection formula gives $p_{!} (p^{*} β \otimes [O (m)]) = β \otimes p_{!} [O (m)]$ , and the standard projective Čech calculation on each affine open $U \subseteq Y$ gives $p_{*} O (m) ∣_{U} = O_{U} [x_{0}, \dots, x_{n}]_{m}$ and $R^{q} p_{*} O (m) ∣_{U} = 0$ for $q > 0$ . The same globally fixed homogeneous monomials of degree $m$ give a basis on every $U$ , and these bases agree on overlaps because the projective-space factor is constant. Thus $p_{*} O (m)$ is the trivial bundle of rank $(\binom{m + n}{n})$ , not merely a bundle with that fibre rank; hence $p_{!} [O (m)] = (\binom{m + n}{n}) [O_{Y}]$ .

2.1L2L3step 1.1algebra

The fibre computation. Let $τ = ch \cdot td$ ; for the generator $α = p_{Y}^{*} β \otimes p_{P}^{*} [O (m)]$ the left-hand side is $τ (p_{!} α) = (\binom{m + n}{n}) ch (β) td (T_{Y})$ by step 1.1 and additivity/multiplicativity of $τ$ on the base [L2]. The right-hand side is $p_{*} (ch (β) ch (O (m)) td (T_{P^{n} \times_{k} Y}))$ ; the tangent bundle of the product is the direct sum of the pullbacks of the tangent bundles, so its Todd class is $td (T_{P^{n}}) td (T_{Y})$ by [L2], and the projection formula for $p_{*}$ [L3] moves $ch (β) td (T_{Y})$ out, leaving the fibre integral $p_{*} (ch (O (m)) td (T_{P^{n}}))$ . The Euler sequence $0 \to O \to O (1)^{n + 1} \to T_{P^{n}} \to 0$ follows on standard charts by differentiating the ratios $x_{i} / x_{j}$ : its first map is the coordinate vector, and these derivatives identify the quotient with the tangent bundle. Consequently $td (T_{P^{n}}) = (h / (1 - e^{- h}))^{n + 1}$ modulo $h^{n + 1}$ , so this fibre integral is the formal residue ${Res}_{h = 0} e^{m h} (1 - e^{- h})^{- n - 1} d h$ , computed by substituting $z = 1 - e^{- h}$ : then $d h = d z / (1 - z)$ and $e^{m h} = (1 - z)^{- m}$ , and the residue equals the coefficient $[z^{n}] (1 - z)^{- m - 1} = (\binom{m + n}{n})$ . Comparing with the left-hand side, the identity holds on every generator.

3.1L1L2L3step 1.1step 2.1algebra∎

General projective bundles, and conclusion. A rank-zero bundle has empty projectivization and both Riemann-Roch sides are zero; assume positive rank for the following computation. The same computation applies to a general projective bundle projection $π : P = P (E) \to Y$ using the relative projective bundle generators of [L1] and the relative Euler sequence: after splitting $E$ by a flag bundle the pushforward coefficient of a relative twist is the divided difference $\sum_{i} F (- x_{i}) / \prod_{j \neq i} (x_{j} - x_{i})$ in the formal roots $x_{i}$ . The relative Euler sequence in the lines convention is $0 \to O \to π^{*} E \otimes O (1) \to T_{P / Y} \to 0$ , from first-order deformations of a line. The pushforward formula follows algebraically from the projective relation $\prod_{i} (h + x_{i}) = 0$ and $π_{*} h^{r - 1} = 1$ : Lagrange interpolation on the nodes $- x_{i}$ reads its top remainder coefficient as that divided difference. The generator integrand is $e^{m h} \prod_{j} (h + x_{j}) / (1 - e^{- x_{j} - h})$ , whose divided-difference sum is the character of the relative symmetric power by the partial-fraction identity $\prod_{j} (1 - z e^{- x_{j}})^{- 1} = \sum_{i} \prod_{j \neq i} (1 - e^{x_{i} - x_{j}})^{- 1} (1 - z e^{- x_{i}})^{- 1}$ ; work initially with independent formal roots and invert their differences to justify interpolation and partial fractions. The resulting coefficients of the symmetric-power series are symmetric polynomials, so denominators cancel; truncation in the Chow grading then specializes correctly even when roots coincide. Thus with injective flag pullback and the base Todd projection formula the identity descends. Since the identity is additive in $α$ and holds on the generators of $K_{0}$ for every $Y$ in the stated hypotheses, the diagram commutes; the case $Y = Spec k$ is Hirzebruch-Riemann-Roch for $P^{n}$ .

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

Riemann-Roch for regular embeddings

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. Let $k$ be an algebraically closed field and $i : X ↪ Y$ a closed immersion of nonsingular irreducible quasi-projective finite type $k$ -varieties, with rank- $d$ normal bundle $N$ . For $E \in K^{0} (X) = G_{0} (X)$ , $ch (i_{*} E) = i_{*} (ch (E) td (N)^{- 1}) in A^{*} (Y)_{Q} .$ Equivalently $ch (i_{*} E) td (T_{Y}) = i_{*} (ch (E) td (T_{X}))$ . The tangent relation is the exact sequence $0 \to T_{X} \to i^{*} T_{Y} \to N \to 0$ , hence $[i^{*} T_{Y}] = [T_{X}] + [N]$ in $K^{0} (X)$ ; a splitting as bundles is neither asserted nor needed. The two $i_{*}$ symbols denote respectively coherent $K$ pushforward and Chow proper pushforward. A vector bundle on $X$ suffices to prove the formula, by finite resolution and additivity.

Facts & Assumptions

Given: the Axiom of Choice; an algebraically closed field $k$ ; a closed immersion $i : X ↪ Y$ of nonsingular irreducible quasi-projective finite type $k$ -varieties with normal bundle $N$ of rank $d$ ; a class $E \in K^{0} (X) = G_{0} (X)$ .

[L1]

The Koszul resolution of a regular section, and exact restriction of finite locally free resolutions of the strict-transform pushforward of a vector bundle to the specified fibres of the deformation blowup; the strict transform and the centre are disjoint (Koszul resolutions and restriction to flat fibres).

[L2]

The deformation to the normal cone of the regular embedding: $b : M \to Y$ proper, with the ordinary fibre $Y$ , the exceptional divisor $P = P_{l i n e s} (N \oplus 1)$ and the strict transform $B = {Bl}_{X} Y$ , and the fibre at infinity $P \cup B$ with intersection $P (N)$ ; the Cartier fibres at $0$ and $\infty$ are linearly equivalent (Deformation to the normal cone and specialization, Smooth immersions, their conormal sequence, deformation charts and smooth sections).

[L3]

Chern character and Todd class are additive and multiplicative in exact sequences, natural under flat pullback, and the Chern classes of a quotient bundle are computed by the Whitney formula; the tangent relation is the exact sequence of Smooth immersions, their conormal sequence, deformation charts and smooth sections (The Chern character and the Todd class, Additivity and multiplicativity of the Chern character and Todd class, Chern classes of a vector bundle on a smooth scheme, Additivity, naturality and the splitting principle for Chern classes).

[L4]

The Chow projection formula for proper pushforward and the identification of the operational action with the ring product (Naturality of the Chow ring and the projection formula, Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes).

Proof

technique · direct; prove the model case for the distinguished section of the projective completion of the normal bundle by Koszul resolution and the regular-section formula, then transport it to the embedding by the deformation blowup and push forward

1.1L1L3L4givenalgebra

Model case. By finite locally free resolution and additivity, it suffices to prove the formula for a vector bundle $E$ ; assume this through the deformation argument. Let $P = P_{l i n e s} (N \oplus 1)$ with projection $p : P \to X$ and universal sequence $0 \to O (- 1) \to p^{*} (N \oplus 1) \to Q \to 0$ , so that the image of the summand $1$ in $Q$ gives a section of $Q$ whose zero scheme is the distinguished section $s : X ↪ P$ , a regular embedding of codimension $d$ and $s^{*} Q = N$ . To check the section hypothesis of [L1], trivialize $N$ over $Spec A$ and use the chart containing $s (X)$ where the last coordinate of the tautological line is $1$ : its generator is $(z_{1}, \dots, z_{d}, 1)$ , and $Q$ has basis the images of the first $d$ standard vectors. The image of the last standard vector has components $- z_{1}, \dots, - z_{d}$ in this basis. Each is a nonzerodivisor after the preceding coordinates are killed in $A [z_{1}, \dots, z_{d}]$ , giving a regular sequence of length $d$ , including the empty sequence when $d = 0$ . Away from $s (X)$ some component is a unit locally, since the line is not the last summand. The Koszul resolution of the regular section of [L1], tensored with $p^{*} E$ , gives $ch (s_{*} E) = ch (p^{*} E) \sum_{j = 0}^{d} (- 1)^{j} ch (⋀^{j} Q^{\lor})$ in $A^{*} (P)_{Q}$ : after injective flag pullback, if $x_{1}, \dots, x_{d}$ are the Chern roots of $Q$ , the alternating sum of exterior powers is $\prod_{j} (1 - e^{- x_{j}}) = c_{d} (Q) td (Q)^{- 1}$ by [L3]. The regular-section formula gives $c_{d} (Q) = s_{*} [X]$ and the ring projection formula gives $c_{d} (Q) γ = s_{*} (s^{*} γ)$ for every $γ \in A^{*} (P)$ ; substituting $γ = td (Q)^{- 1} ch (p^{*} E)$ gives $ch (s_{*} E) = s_{*} (td (N)^{- 1} ch (E))$ , the asserted formula in the model case.

1.2L1L2L3givenalgebra

Reduction to the model. Let $b : M \to Y$ be the deformation blowup of [L2] with $F : X \times P^{1} ↪ M$ the strict transform, $j_{0} : Y ↪ M$ the ordinary fibre, $k : P ↪ M$ the fibre $P = P (N \oplus 1)$ and $l : B ↪ M$ the strict transform of $Y \times {\infty}$ . Resolve the sheaf $F_{*} {pr}_{X}^{*} E$ by a bounded complex $G_{∙}$ of vector bundles and put $γ = \sum_{j} (- 1)^{j} ch (G_{j})$ , an operational class on $M$ . By [L1] this complex restricts on $j_{0}$ to a resolution of $i_{*} E$ , on $k$ to a resolution of $s_{*} E$ , and is exact on $l$ . Their Chern characters restrict along these smooth embeddings because they are polynomials in Chern classes, whose naturality holds for arbitrary smooth-scheme morphisms. The two Cartier fibres at $0$ and $\infty$ are linearly equivalent, so the cycle identity $j_{0 *} [Y] = k_{*} [P] + l_{*} [B]$ holds in $A_{*} (M)$ with multiplicity one because the centre is smooth; capping with $γ$ and using the Chern projection formula gives $j_{0 *} (ch (i_{*} E) \cap [Y]) = k_{*} (ch (s_{*} E) \cap [P])$ , the $B$ -term vanishing because the restricted complex is exact.

2.1L2L3L4step 1.1step 1.2algebra∎

Conclusion. Push the equality of step 1.2 forward along the proper morphism $b$ : since $b j_{0} = {id}_{Y}$ and $b k = i p$ , and proper pushforward is functorial with the projection formula [L4], one gets $ch (i_{*} E) \cap [Y] = i_{*} p_{*} (ch (s_{*} E) \cap [P]) = i_{*} (ch (E) td (N)^{- 1} \cap [X])$ by the model computation of step 1.1 and $p s = {id}_{X}$ . Hence $ch (i_{*} E) = i_{*} (ch (E) td (N)^{- 1})$ ; the equivalent Todd-tangent form follows from the exact tangent sequence $0 \to T_{X} \to i^{*} T_{Y} \to N \to 0$ and the multiplicativity of the Todd class, using $[i^{*} T_{Y}] = [T_{X}] + [N]$ and no bundle splitting. The finite locally free resolution extends the identity from vector bundles to all classes of $K_{0} (X)$ by additivity.

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

Grothendieck-Riemann-Roch for projective morphisms

Statement

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. Let $k$ be an algebraically closed field and let $f : X \to Y$ be a projective morphism of nonsingular irreducible quasi-projective $k$ -schemes of finite type (Projective morphisms before Proj, Smooth morphism of schemes, Proper morphisms); projectivity is exactly the hypothesis that $f$ factors as a closed immersion $i : X ↪ P_{k}^{n} \times_{k} Y$ followed by the projection $p : P_{k}^{n} \times_{k} Y \to Y$ for some $n \geq 0$ (A projective morphism has a relative Proj presentation). Then for every $α \in K_{0} (X)$ (Grothendieck groups of coherent sheaves and of vector bundles on a scheme, Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes): $ch (f_{!} (α)) \cdot td (T_{Y}) = f_{*} (ch (α) \cdot td (T_{X})) in A^{*} (Y)_{Q},$ where $ch, td$ are the Chern character and Todd class of The Chern character and the Todd class, $f_{!}$ is the $K$ -theory pushforward (Pushforward of coherent sheaves in algebraic K-theory) and $f_{*}$ is the proper Chow pushforward (Proper pushforward of cycles and the norm formula). Equivalently $τ_{Y} \circ f_{!} = f_{*} \circ τ_{X}$ for $τ_{S} (α) = ch (α) td (T_{S})$ . For $Y = Spec k$ this is Hirzebruch-Riemann-Roch $χ (X, F) = \deg (ch (F) td (T_{X}))$ (Euler characteristic of a coherent sheaf).

Facts & Assumptions

Given: the Axiom of Choice; an algebraically closed field $k$ ; a projective morphism $f : X \to Y$ of nonsingular irreducible quasi-projective finite type $k$ -schemes; a factorization $f = p \circ i$ with $i : X ↪ P_{k}^{n} \times_{k} Y$ a closed immersion and $p$ the projection.

[L1]

Riemann-Roch holds for the closed immersion $i$ : $ch (i_{*} E) = i_{*} (ch (E) td (N)^{- 1})$ , equivalently $τ \circ i_{*} = i_{*} \circ τ$ (Riemann-Roch for regular embeddings).

[L2]

Riemann-Roch holds for the projection $p$ : $τ_{Y} \circ p_{!} = p_{*} \circ τ_{P^{n} \times_{k} Y}$ (Riemann-Roch for projective-space projections).

[L3]

$K$ -theory pushforward is functorial under composition, $(p \circ i)_{!} = p_{!} \circ i_{!}$ , and Chow proper pushforward is functorial, $(p \circ i)_{*} = p_{*} i_{*}$ ; $K_{0}$ on the smooth quasi-projective source is generated by vector bundles (Pushforward of coherent sheaves in algebraic K-theory, Proper pushforward of cycles and the norm formula, Vector-bundle K-theory equals coherent K-theory on regular quasi-projective schemes).

[L4]

The projection formula and naturality of the Chern character and Todd class for flat morphisms identify the two composites; the target of $τ$ is the codimension-graded Chow group of a smooth scheme (Naturality of the Chow ring and the projection formula, Additivity and multiplicativity of the Chern character and Todd class, Projection formula for higher direct images and K-theory pushforward).

Proof

technique · direct; factor the projective morphism into a closed immersion followed by a projective-space projection and compose the two Riemann-Roch squares

1.1L3givenalgebra

The factorization. By the relative Proj presentation of a projective morphism, $f$ factors as $f = p \circ i$ with $i : X ↪ P_{k}^{n} \times_{k} Y$ a closed immersion of nonsingular varieties and $p$ the projection; both factors are morphisms of nonsingular irreducible quasi-projective $k$ -schemes, and $p$ is the projection to the second factor.

2.1L1L2L3step 1.1algebra

Composition of the two squares. Consider the two commutative squares: $τ \circ i_{*} = i_{*} \circ τ$ on $X \to P^{n} \times_{k} Y$ by [L1], and $τ \circ p_{!} = p_{*} \circ τ$ on $P^{n} \times_{k} Y \to Y$ by [L2]. Composing them gives $τ_{Y} \circ p_{!} \circ i_{!} = p_{*} \circ i_{*} \circ τ_{X}$ , and by the functoriality of both pushforwards, $p_{!} \circ i_{!} = f_{!}$ and $p_{*} i_{*} = f_{*}$ , this is exactly $τ_{Y} \circ f_{!} = f_{*} \circ τ_{X}$ , the displayed formula.

3.1L3L4step 2.1givenalgebra∎

Coherent classes and Hirzebruch-Riemann-Roch. Since $X$ is smooth and quasi-projective, $K_{0} (X) = K^{0} (X)$ and the identity proved for vector bundles extends to all coherent classes by additivity of $ch$ , of $f_{!}$ and of $f_{*}$ along finite resolutions [L3]. For $Y = Spec k$ , the Chow pushforward is the degree and the $K$ -theory pushforward is the Euler characteristic, so the formula becomes Hirzebruch-Riemann-Roch $χ (X, F) = \deg (ch (F) td (T_{X}))$ ; the transport of $ch$ and $td$ to coherent classes uses the naturality and multiplicativity of [L4].

RemarkRemark: Literature-sourcedProof: Not applicableOpen item page →

Conventions for the Chow ring and Grothendieck-Riemann-Roch

Conventions

Assume the Axiom of Choice (The Axiom of Choice) inherited from the finite coherent-resolution and cohomological suppliers. (i) Rational equivalence is the version of Rational equivalence and the Chow group of cycles (Stacks 42.19.1, tag 02RW; Fulton 1.3), with order functions on possibly singular integral closed subschemes supplied by The order function of a one-dimensional Noetherian local domain; the equivalent $P^{1}$ -parametrized definition (Fulton 1.6; Stacks 43.8-43.9) is not used but is consistent with it. (ii) $A^{*} (X)$ always denotes the codimension-graded intersection ring of a smooth equidimensional $k$ -scheme (The intersection product and Chow ring of a smooth scheme); on general schemes $A_{*} (X)$ and operational Chern cap maps are used; ring-valued Chern classes, character and Todd classes are obtained through the operational identification on smooth schemes. (iii) The intersection product is the diagonal Gysin construction (Stacks 42.62, tag 0FC0); the moving-lemma/Serre Tor-formula construction of Stacks Chapter 43 is an independent route to the same product for nonsingular projective varieties over an algebraically closed field and is not needed here. (iv) In $K$ -theory, $K_{0}$ and $K^{0}$ are identified on regular quasi-projective finite-type schemes (Grothendieck groups of coherent sheaves and of vector bundles on a scheme), and pushforward is defined via higher direct images with the AC/DC inheritance declared in Pushforward of coherent sheaves in algebraic K-theory.

Hypotheses of the main theorem

Grothendieck-Riemann-Roch for projective morphisms is stated for an algebraically closed base field $k$ , nonsingular irreducible quasi-projective $k$ -schemes $X, Y$ , and a projective (hence proper) morphism $f$ ; the field hypothesis agrees with the Borel-Serre proof scope. Vakil Classes 14/16/17/19 are partial comparisons with omitted details; the complete proof used here is the current local supplier chain, with the Borel-Serre introduction and Sections 7-16 as the independently retrieved comparison. The design-named Fulton source entry was dropped by the owner with confidence certain after five item-level alternatives per page were recorded; Fulton was not retrieved or read and is retained only as a bibliographical comparison, never as a proof premise. No claim is made here for singular targets, for proper non-projective morphisms, for pairs over a non-algebraically-closed field, or in the $K$ -theory of perfect complexes; the singular and bivariant versions (Fulton, Intersection Theory, Chapters 18 and 17) are outside this pair.

Choice

The Axiom of Choice (and, where the coherence of higher direct images is invoked, Dependent Choice) is inherited from the published coherent-cohomology suppliers throughout; it is used to form the resolutions and direct images in Pushforward of coherent sheaves in algebraic K-theory and is recorded in Grothendieck groups of coherent sheaves and of vector bundles on a scheme. The order function and the finiteness of the lengths it computes also use the Axiom of Choice, as recorded in The order function of a one-dimensional Noetherian local domain, so the whole Chow-theoretic chain of this page carries the assumption explicitly. No choice-free claim is made stronger here; the definitional part of the Chow groups uses no choice beyond the free abelian group on a set, as recorded in Algebraic cycles and the cycle group of a scheme of finite type over a field.

5 · Examples, counterexamples and false statements

None yet.