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Internal Hom from a finitely presented sheaf is quasi-coherent

Statement

Assume the Axiom of Choice, inherited from the affine equivalence (The Axiom of Choice). Let $X$ be a scheme, let $F$ be a finitely presented quasi-coherent $O_{X}$ -module and let $G$ be a quasi-coherent $O_{X}$ -module (Finite type and finitely presented module sheaves, Quasi-coherent module on a scheme). Write $H o m_{O_{X}} (F, G)$ for the internal Hom, the sheaf $U \mapsto {Hom}_{O_{U}} (F ∣_{U}, G ∣_{U})$ (Internal Hom of module sheaves).

Then:

$H o m_{O_{X}} (F, G)$ is quasi-coherent.
If $U = Spec A$ is an affine open with $F ∣_{U} ≅ \tilde{M}$ , $G ∣_{U} ≅ \tilde{N}$ for $A$ -modules $M, N$ with $M$ finitely presented, then there is a canonical isomorphism of $O_{U}$ -modules $H o m_{O_{X}} (F, G) ∣_{U} ≅ \tilde{{Hom}_{A} (M, N)},$ natural in $M$ and $N$ ; on $D (f) \subseteq U$ it identifies sections with ${Hom}_{A_{f}} (M_{f}, N_{f})$ via the natural localisation map ${Hom}_{A} (M, N)_{f} \to {Hom}_{A_{f}} (M_{f}, N_{f})$ , which is an isomorphism because $M$ is finitely presented.
The identifications of (2) are compatible on overlaps of affine charts, both sides being given by restriction of morphisms.

The finite presentation hypothesis on $F$ is used exactly through the isomorphism of (2); for arbitrary quasi-coherent $F$ no such conclusion is asserted.

Facts & Assumptions

Given: The Axiom of Choice; a scheme $X$ ; a finitely presented quasi-coherent $F$ ; a quasi-coherent $G$ .

[F1]

Sections of the internal Hom are Hom modules: for open $U$ , $Γ (U, H o m (F, G)) = {Hom}_{O_{U}} (F ∣_{U}, G ∣_{U})$ , with restriction of a morphism as restriction map; this definition makes no quasi-coherence claim (Internal Hom of module sheaves).

[F2]

Affine equivalence: for $X = Spec A$ the functor $M \mapsto \tilde{M}$ is an equivalence onto the quasi-coherent $O_{X}$ -modules, with $M ≅ Γ (X, \tilde{M})$ , and for all $A$ -modules $M, N$ the map $u \mapsto \tilde{u}$ is a bijection ${Hom}_{A} (M, N) \to {Hom}_{O_{X}} (\tilde{M}, \tilde{N})$ (Affine quasi-coherent sheaves are modules).

[F3]

Localisation of Hom: for a multiplicative subset $S \subseteq A$ and $A$ -modules $M, N$ the natural map $S^{- 1} {Hom}_{A} (M, N) \to {Hom}_{S^{- 1} A} (S^{- 1} M, S^{- 1} N)$ is injective for $M$ finitely generated and an isomorphism for $M$ finitely presented (Localisation of Hom for finite and finitely presented modules).

[F4]

Restrictions of associated sheaves: for $D (f) \subseteq Spec A$ one has $(\tilde{M}) ∣_{D (f)} = \tilde{(M_{f})}$ , and sections on $D (f)$ are $M_{f}$ with localisation as restriction (An associated sheaf restricts to an associated sheaf on an affine open, Module sheaf on an affine scheme, The associated module sheaf exists).

[F5]

The conditions in [F1]–[F4] are local: finite presentation and quasi-coherence of $F$ and quasi-coherence of $G$ hold on some affine open neighbourhood of every point, and on a distinguished open of an affine chart the modules remain finitely presented respectively arbitrary (Finite type and finitely presented module sheaves, Quasi-coherent module on a scheme).

Proof technique: direct; compute the distinguished-open sections of the internal Hom on an affine chart, identify them with the localisations of ${Hom}_{A} (M, N)$ by the localisation-of-Hom theorem, and use the basis-determination of associated sheaves.

Proof

1.1F2F5

An affine chart where both sheaves are associated: let $x \in X$ . By [F5] and the affine equivalence [F2] applied to an affine neighbourhood of $x$ , and shrinking to a distinguished open inside the intersection of a chart for $F$ and a chart for $G$ , there is an affine open $U = Spec A$ containing $x$ with $F ∣_{U} ≅ \tilde{M}$ and $G ∣_{U} ≅ \tilde{N}$ , where $M$ is finitely presented and $N$ is an arbitrary $A$ -module; such charts cover $X$ .

2.1F1F2F4step 1.1

Sections on distinguished opens: for $f \in A$ , [F1] gives $Γ (D (f), H o m (F, G)) = {Hom}_{O_{D (f)}} (F ∣_{D (f)}, G ∣_{D (f)})$ , and by [F4] $F ∣_{D (f)} = \tilde{(M_{f})}$ and $G ∣_{D (f)} = \tilde{(N_{f})}$ ; since $D (f)$ is affine, [F2] identifies this Hom module canonically with ${Hom}_{A_{f}} (M_{f}, N_{f})$ .

3.1F1F2F3step 2.1

The restriction maps are the localisation maps: for $D (g) \subseteq D (f)$ the restriction of a morphism of $O_{U}$ -modules is the map $ψ \mapsto ψ ∣_{D (g)}$ , so under the identifications of step 2.1 the restriction $Γ (D (f), H o m) \to Γ (D (g), H o m)$ corresponds to ${Hom}_{A} (M, N) \to {Hom}_{A_{f}} (M_{f}, N_{f}) \to {Hom}_{A_{g}} (M_{g}, N_{g})$ obtained by functoriality of restriction of morphisms, which is exactly the composite of the natural localisation maps for $Hom$ ; by [F3] this composite is the canonical localisation ${Hom}_{A} (M, N)_{f} \to {Hom}_{A} (M, N)_{g}$ under the identifications ${Hom}_{A} (M, N)_{f} ≅ {Hom}_{A_{f}} (M_{f}, N_{f})$ and ${Hom}_{A} (M, N)_{g} ≅ {Hom}_{A_{g}} (M_{g}, N_{g})$ , both isomorphisms because $M$ is finitely presented.

4.1F2F3F4step 2.1step 3.1

The identification with the associated sheaf: by steps 2.1 and 3.1 the distinguished-open data of $H o m (F, G) ∣_{U}$ are the modules ${Hom}_{A} (M, N)_{f}$ , with restrictions the localisation maps, which are exactly the distinguished-open data of the associated sheaf $\tilde{{Hom}_{A} (M, N)}$ ; a morphism of $O_{U}$ -modules between these two sheaves is determined by its components on distinguished opens and a compatible family of isomorphisms on the basis extends uniquely, so the identity family assembles to a canonical isomorphism $H o m (F, G) ∣_{U} ≅ \tilde{{Hom}_{A} (M, N)}$ , natural in $M$ and $N$ because all identifications used are the canonical ones of [F1]–[F4]. This proves (2).

5.1F1step 4.1

Claim 1 and claim 3: the charts $U = Spec A$ of step 1.1 cover $X$ and on each of them $H o m (F, G) ∣_{U} ≅ \tilde{{Hom}_{A} (M, N)}$ is associated, so by the local form of the definition $H o m (F, G)$ is quasi-coherent, proving claim 1. On an overlap $U \cap U^{'}$ of two such charts both isomorphisms are built from the restriction-of-morphisms identifications of the same sheaf $H o m (F, G)$ , so they agree on the overlap: this is claim 3, and the canonical identifications restrict correctly on distinguished opens.

6.1F2F3step 4.1∎

Choice accounting: no choice is used beyond the inherited Axiom of Choice of the affine equivalence [F2], which is used to identify Hom modules with Hom of associated sheaves; the localisation-of-Hom isomorphisms of [F3] are canonical, and all selections in step 1.1 involve finitely many charts around one point at a time.

Depends on

Used by

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Sources

The Stacks Project, Schemes, §§26.5, 26.7, 26.24 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Chapters 6, 14, 17 (standard reference, not scraped)