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Projective bundle represents line quotients

Statement

Assume the Axiom of Choice as inherited from the relative Proj and sheaf constructions (The Axiom of Choice). Let $S$ be a scheme, let $E$ be a finite locally free $O_{S}$ -module of locally constant rank (Locally free sheaves of finite rank), and let $π : P_{S} (E) = {Proj}_{S} Sym (E) ⟶ S$ be its projective bundle in the quotient convention, with twist $O_{P_{S} (E)} (1)$ and tautological quotient $π^{*} E \to O_{P_{S} (E)} (1)$ (Projective bundle in the quotient convention, Relative Proj of a graded quasi-coherent algebra).

Then for every $S$ -scheme $g : T \to S$ the assignment $h ⟼ (h^{*} (π^{*} E \to O_{P_{S} (E)} (1)))$ is a natural bijection between

the set of $S$ -morphisms $h : T \to P_{S} (E)$ , and
the set of isomorphism classes of surjections $g^{*} E \to L$ with $L$ an invertible $O_{T}$ -module (Invertible sheaves), where an isomorphism between $g^{*} E \to L$ and $g^{*} E \to L^{'}$ is an isomorphism $L \to L^{'}$ of $O_{T}$ -modules making the triangle commute.

Naturality means compatibility with morphisms $T^{'} \to T$ of $S$ -schemes. For $T = P_{S} (E)$ and $h = id$ the class on the right is the tautological quotient itself: ${id}^{*} (π^{*} E \to O (1))$ is $π^{*} E \to O (1)$ . The rank-zero case is included: if $E = 0$ then $P_{S} (0) = \emptyset$ , and the two sides are empty for nonempty $T$ and singletons for $T = \emptyset$ .

Facts & Assumptions

Given: A scheme $S$ , a finite locally free $O_{S}$ -module $E$ of locally constant rank, the projective bundle $π : P_{S} (E) \to S$ with twist $O (1) = O_{P_{S} (E)} (1)$ and tautological quotient $q : π^{*} E \to O (1)$ , an $S$ -scheme $g : T \to S$ , and the Axiom of Choice as inherited from the relative Proj construction.

[A1]

The Axiom of Choice states that every family of nonempty sets has a choice function. (The Axiom of Choice)

[F1]

For an affine open $U \subseteq S$ over which $E ∣_{U} ≅ O_{U}^{r}$ with $r \geq 1$ : the restriction $P_{S} (E) ∣_{U} = π^{- 1} (U)$ equals $P_{U} (E ∣_{U})$ (Relative Proj of a graded quasi-coherent algebra), there is a canonical isomorphism $P_{S} (E) \times_{S} U ≅ P_{U}^{r - 1}$ under which $Sym (E) ∣_{U}$ corresponds to $O_{U} [t_{1}, \dots, t_{r}]$ with degree-one generators, the twist $O (1) ∣_{π^{- 1} (U)}$ corresponds to the standard twist $O_{P_{U}^{r - 1}} (1)$ , and the tautological quotient restricts to the standard quotient $O_{U}^{r} \to O_{P_{U}^{r - 1}} (1)$ whose components are the coordinate sections. If $E ∣_{U} = 0$ then $P_{U} (0) = \emptyset$ . The twist $O (1)$ is invertible on $π^{- 1} (U)$ in the case $r \geq 1$ . (Projective bundle in the quotient convention, Relative Proj of a graded quasi-coherent algebra, Relative projective space from standard charts, Invertible twists for degree-one generated rings)

[F2]

A finite locally free module of locally constant rank admits an open cover of $S$ by affine opens $U$ with $E ∣_{U} ≅ O_{U}^{r}$ for a constant $r = r (U)$ ; its rank function is locally constant. Consequently such a cover by trivialising affine opens exists, and the rank attached to a connected trivialising open set is well defined. (Locally free sheaves of finite rank)

[F3]

Pullback of $O$ -modules (Pullback of a module along a morphism of ringed spaces): ${id}^{*} = id$ and $(h \circ u)^{*} = u^{*} h^{*}$ for composable morphisms; if $G ∣_{V} ≅ O_{V}^{r}$ then $h^{*} G ∣_{h^{- 1} (V)} ≅ O_{h^{- 1} (V)}^{r}$ , so pullback of a locally free sheaf of rank $r$ is locally free of rank $r$ and pullback of an invertible sheaf is invertible; in particular $π^{*} E$ , $g^{*} E$ and $h^{*} O (1)$ are locally free of the corresponding ranks on their sources. (Pullback of a module along a morphism of ringed spaces, Locally free sheaves of finite rank, Invertible sheaves)

[F4]

For an invertible sheaf $L$ with global sections $s_{0}, \dots, s_{n}$ , the associated morphism $O_{X}^{n + 1} \to L$ , $(a_{j}) \mapsto \sum_{j} a_{j} s_{j}$ , is surjective exactly when the $s_{j}$ generate $L$ , and a morphism from a free module is determined by its components; in particular a surjection $O_{X}^{n + 1} \to L$ is the same thing as a generating $(n + 1)$ -tuple, namely the images of the standard basis. If the $s_{j}$ generate $L$ and $u : Y \to X$ is a morphism then the pullbacks $u^{*} s_{j}$ generate $u^{*} L$ , so the pullback of a surjection of invertible sheaves is again surjective. (Global generation by the evaluation map, Invertible sheaves, Pullback of a module along a morphism of ringed spaces)

[F5]

For a base scheme $S$ and $n \geq 0$ , the assignment $φ \mapsto (φ^{*} O (1); φ^{*} x_{0}, \dots, φ^{*} x_{n})$ is a natural bijection between $S$ -morphisms $φ : T \to P_{S}^{n}$ and isomorphism classes of pairs $(L; s_{0}, \dots, s_{n})$ with $L$ invertible and $s_{0}, \dots, s_{n}$ generating $L$ ; the morphism attached to data satisfies $φ^{*} O (1) ≅ L$ and $φ^{- 1} (D_{+} (x_{i})) = X_{s_{i}}$ , and a morphism is determined by its data. In particular, surjections $O_{T}^{n + 1} \to L$ correspond bijectively to morphisms $T \to P_{S}^{n}$ by sending the surjection $q$ to the morphism attached to the generating tuple $(q (e_{0}), \dots, q (e_{n}))$ of images of the standard basis $e_{0}, \dots, e_{n}$ of $O_{T}^{n + 1}$ . (Maps to projective space equal generating line-bundle data, Generating line-bundle sections define a morphism to projective space, Global generation by the evaluation map)

[F6]

Morphisms of schemes are local on the source: two morphisms agreeing on the members of an open cover are equal, and a compatible family of morphisms on an open cover glues uniquely. Morphisms of sheaves and invertible sheaves are likewise local: compatible local data, including the overlap identifications, glue, and a morphism of sheaves is determined by its local restrictions. If $q : M \to L$ is a surjection of $O$ -modules and $α, β : L \to L^{'}$ satisfy $α \circ q = β \circ q$ , then $α = β$ , because the image of $q$ generates $L$ locally and a morphism of sheaves is determined by its values on a generating family of local sections. (Morphisms of schemes are local on compatible open covers, Compatible local sheaves glue uniquely up to unique isomorphism, A sheaf on a topological space, Modules on a ringed space)

Proof

technique · direct: define the restriction map by pulling back the universal quotient; trivialise the bundle over affine opens of $S$, where the projective bundle becomes a projective space with the standard quotient and the absolute line-bundle-data equivalence gives the bijection; then glue the local bijections over the cover of $T$ induced by the trivialising cover of $S$, using uniqueness of morphisms and of isomorphisms conjugating a surjection

1.1A1F1F2

The trivialising cover and the local models. By [F2] and the Axiom of Choice [A1] choose an open cover $S = ⋃_{i \in I} U_{i}$ by affine opens with trivialisations $τ_{i} : O_{U_{i}}^{r_{i}} \to E ∣_{U_{i}}$ , $r_{i} \geq 0$ . For an $S$ -scheme $h : T \to S$ put $T_{i} = h^{- 1} (U_{i})$ and $P_{i} = π^{- 1} (U_{i}) = P_{U_{i}} (E ∣_{U_{i}})$ , so that the $T_{i}$ cover $T$ and the $P_{i}$ cover $P_{S} (E)$ . By [F1], if $r_{i} \geq 1$ there is an isomorphism $P_{i} ≅ P_{U_{i}}^{r_{i} - 1}$ identifying $O (1) ∣_{P_{i}}$ with the standard invertible twist and $q ∣_{P_{i}}$ with the standard quotient $O_{U_{i}}^{r_{i}} \to O (1)$ with components the coordinate sections $x_{0}, \dots, x_{r_{i} - 1}$ , while if $r_{i} = 0$ then $P_{i} = \emptyset$ .

1.2F1F3F4algebra

The map and its naturality. For $h : T \to P_{S} (E)$ the pullback $h^{*} q : h^{*} π^{*} E \to h^{*} O (1)$ is a morphism $g^{*} E \to h^{*} O (1)$ , because $h^{*} π^{*} E = (π \circ h)^{*} E = g^{*} E$ by [F3]; the target $h^{*} O (1)$ is invertible by [F3] applied to the invertible sheaf $O (1)$ , and $h^{*} q$ is surjective: on $T_{i} = h^{- 1} (U_{i})$ with $r_{i} \geq 1$ it is the pullback of the standard quotient of [F1], whose components are the pullbacks of the generating coordinate sections $x_{j}$ of $O (1)$ on $P_{U_{i}}^{r_{i} - 1}$ (each $x_{j}$ is a frame on the chart $D_{+} (x_{j})$ ), and pullbacks of generating sections generate, so $h^{*} q ∣_{T_{i}}$ is surjective by [F4]; when $r_{i} = 0$ one has $P_{i} = π^{- 1} (U_{i}) = \emptyset$ by [F1] while $π \circ h = g$ forces $h (T_{i}) \subseteq P_{i}$ , hence $T_{i} = \emptyset$ and there is nothing to check. Thus $Φ_{T} (h) : = [h^{*} q]$ is an isomorphism class of surjections $g^{*} E \to L$ with $L$ invertible, and for a morphism $u : T^{'} \to T$ of $S$ -schemes one has $Φ_{T} (h \circ u) = u^{*} Φ_{T} (h)$ since $(h \circ u)^{*} = u^{*} h^{*}$ by [F3].

2.1F1F4F5step 1.1algebra

The local bijection for $r \geq 1$ . Fix $i$ with $r_{i} = r \geq 1$ and fix $h : T \to S$ ; write $g_{i} = g ∣_{T_{i}}$ and identify $E ∣_{T_{i}} ≅ O_{T_{i}}^{r}$ by $τ_{i}$ . By [F5] applied over the base $U_{i}$ , the $U_{i}$ -morphisms $φ : T_{i} \to P_{U_{i}}^{r - 1}$ correspond bijectively to generating tuples $(s_{0}, \dots, s_{r - 1})$ of an invertible sheaf $L$ on $T_{i}$ , by $φ^{*} x_{j} \leftrightarrow s_{j}$ , and such tuples correspond bijectively to surjections $g_{i}^{*} E ≅ O_{T_{i}}^{r} \to L$ by $s_{j} = q (e_{j})$ for the standard basis, a morphism from a free module being determined by its components and surjective exactly when the components generate by [F4]. Under the identification of [F1] the pullback of the universal quotient $q$ along a morphism is the pullback of the standard quotient, whose components are the pullbacks of the coordinate sections; hence $Φ_{T_{i}} (φ)$ corresponds under these bijections to the tuple $(φ^{*} x_{j}) = (s_{j})$ , i.e. $Φ$ followed by the two bijections is the identity. Therefore $Φ_{T_{i}}$ is a bijection between $U_{i}$ -morphisms $T_{i} \to P_{i}$ and isomorphism classes of surjections $g_{i}^{*} E \to L_{i}$ with $L_{i}$ invertible on $T_{i}$ , the local model being that of step 1.1.

2.2F1F3step 1.1algebra

The local bijection for $r = 0$ . If $r_{i} = 0$ then $P_{i} = \emptyset$ by [F1], so a morphism $T_{i} \to P_{i}$ exists only when $T_{i} = \emptyset$ ; and a surjection $g_{i}^{*} E = 0 \to L_{i}$ onto an invertible sheaf $L_{i}$ exists only when $T_{i} = \emptyset$ , since an invertible sheaf on a nonempty scheme has nonzero stalks (Invertible sheaves) while the zero sheaf does not, and the zero morphism onto such a sheaf is not surjective. For $T_{i} = \emptyset$ both sides have exactly one element, the empty morphism and the zero surjection of the empty scheme; hence $Φ_{T_{i}}$ is a bijection here as well.

3.1F6step 2.1step 2.2

Injectivity of $Φ_{T}$ . Let $h_{1}, h_{2} : T \to P_{S} (E)$ with $Φ_{T} (h_{1}) ≅ Φ_{T} (h_{2})$ . For each $i$ the restrictions are isomorphic: $Φ_{T_{i}} (h_{1} ∣_{T_{i}}) = Φ_{T} (h_{1}) ∣_{T_{i}} ≅ Φ_{T} (h_{2}) ∣_{T_{i}} = Φ_{T_{i}} (h_{2} ∣_{T_{i}})$ , because pullback of the universal quotient commutes with restriction to the open subscheme $T_{i}$ . By the bijectivity of steps 2.1 and 2.2 the restrictions $h_{1} ∣_{T_{i}}$ and $h_{2} ∣_{T_{i}}$ are equal for every $i$ , and since the $T_{i}$ cover $T$ the morphisms $h_{1}, h_{2}$ are equal by [F6]. Hence $Φ_{T}$ is injective.

3.2F5step 2.1step 2.2

Surjectivity of $Φ_{T}$ , local construction. Let $q^{'} : g^{*} E \to L$ be a surjection with $L$ invertible on $T$ ; we construct $h : T \to P_{S} (E)$ with $Φ_{T} (h) ≅ q^{'}$ . For each $i$ : if $r_{i} \geq 1$ , let $h_{i} : T_{i} \to P_{i}$ be the morphism corresponding to the isomorphism class of $q^{'} ∣_{T_{i}}$ under the bijection of step 2.1 (equivalently, the morphism attached by [F5] to the generating tuple $(q^{'} (τ_{i} (e_{j})))_{j}$ ); then $Φ_{T_{i}} (h_{i}) ≅ q^{'} ∣_{T_{i}}$ . If $r_{i} = 0$ then $T_{i} = \emptyset$ , as the restriction of $q^{'}$ would be a surjection $0 \to L ∣_{T_{i}}$ onto an invertible sheaf, which is impossible on a nonempty $T_{i}$ by step 2.2; take $h_{i}$ the empty morphism.

4.1F6step 2.1step 3.1step 3.2

The local morphisms glue. Let $i, j$ ; on $T_{i j} = T_{i} \cap T_{j}$ the restrictions $h_{i} ∣_{T_{i j}}$ and $h_{j} ∣_{T_{i j}}$ are morphisms $T_{i j} \to P_{i} ∣_{T_{i j}} = P_{j} ∣_{T_{i j}}$ with $Φ_{T_{i j}} (h_{i} ∣_{T_{i j}}) ≅ q^{'} ∣_{T_{i j}} ≅ Φ_{T_{i j}} (h_{j} ∣_{T_{i j}})$ , the first identification being the restriction of the isomorphism of step 3.2 and the second the same statement with $j$ in place of $i$ . Since $Φ_{T_{i j}}$ is injective by step 3.1 (or by the bijection of step 2.1 applied over the open subscheme $T_{i j}$ of $T_{i}$ ), the two restrictions are equal; the open subschemes $T_{i j}$ cover $T_{i} \cap T_{j}$ (they equal it when nonempty, and when $T_{i j} = \emptyset$ there is nothing to check). Hence the family $(h_{i})$ is compatible on the cover ${T_{i}}$ of $T$ and glues to a unique morphism $h : T \to P_{S} (E)$ by [F6].

5.1F6step 3.2step 4.1

The glued morphism represents $q^{'}$ . For each $i$ one has $Φ_{T} (h) ∣_{T_{i}} = Φ_{T_{i}} (h ∣_{T_{i}}) = Φ_{T_{i}} (h_{i}) ≅ q^{'} ∣_{T_{i}}$ , so there is an isomorphism $α_{i} : h^{*} O (1) ∣_{T_{i}} \to L ∣_{T_{i}}$ with $α_{i} \circ (h^{*} q) ∣_{T_{i}} = q^{'} ∣_{T_{i}}$ . On an overlap $T_{i j}$ the two isomorphisms $α_{i} ∣_{T_{i j}}$ and $α_{j} ∣_{T_{i j}}$ both conjugate the surjection $(h^{*} q) ∣_{T_{i j}}$ to $q^{'} ∣_{T_{i j}}$ ; such a conjugating isomorphism is unique by the last clause of [F6], since $(h^{*} q) ∣_{T_{i j}}$ is surjective. Hence $α_{i}$ and $α_{j}$ agree on overlaps, so by the gluing clause of [F6] they glue to an isomorphism $α : h^{*} O (1) \to L$ satisfying $α \circ h^{*} q = q^{'}$ ; that is, $Φ_{T} (h) ≅ q^{'}$ . Therefore $Φ_{T}$ is surjective.

6.1

Conclusion. Steps 1.2, 3.1 and 5.1 show that $Φ_{T}$ is a natural bijection for every $S$ -scheme $g : T \to S$ , and steps 2.1 and 2.2 supply the local bijections it is built from. The tautological quotient is the universal element: for $T = P_{S} (E)$ and $h = id$ one has $Φ_{T} (id) = {id}^{*} q = q$ by [F3]. Naturality in the base holds because the relative Proj and $Sym$ commute with base change $S^{'} \to S$ , so $P_{S^{'}} (E_{S^{'}}) ≅ P_{S} (E) \times_{S} S^{'}$ and the universal quotient base changes to the universal quotient (Projective bundle in the quotient convention). In the rank-zero case $E = 0$ one has $P_{S} (0) = \emptyset$ ; for $T \neq \emptyset$ there is no morphism $T \to \emptyset$ and no surjection $0 \to L$ onto an invertible sheaf, while for $T = \emptyset$ both sides consist of the empty morphism and the zero surjection. The Axiom of Choice [A1] is inherited from the relative Proj construction and is used to select a trivialising affine cover of $S$ in step 1.1; every later step is determined by that finite-or-infinite family of local data, with no further choices. [A1, F3, step 1.2, step 2.1, step 2.2, step 3.1, step 5.1, cases: rank zero and empty T] \qed

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Projective bundle of a trivial module Example

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Sources

The Stacks Project, Constructions of Schemes, Sections 27.8-27.21 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Sections 4.5, 7.4, 9.3, 10.6, 17.4, 17.6, 18.2 (standard reference, not scraped)