Lemma 26.7.1 (01I7)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 26: Schemes
Section 26.7: Quasi-coherent sheaves on affines
Lemma 26.7.1 (cite)

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Lemma 26.7.1. Let $(X, \mathcal{O}_ X) = (\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)})$ be an affine scheme. Let $M$ be an $R$ -module. There exists a canonical isomorphism between the sheaf $\widetilde M$ associated to the $R$ -module $M$ (Definition 26.5.3) and the sheaf $\mathcal{F}_ M$ associated to the $R$ -module $M$ (Modules, Definition 17.10.6). This isomorphism is functorial in $M$ . In particular, the sheaves $\widetilde M$ are quasi-coherent. Moreover, they are characterized by the following mapping property

$\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\widetilde M, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _ R(M, \Gamma (X, \mathcal{F}))$

for any sheaf of $\mathcal{O}_ X$ -modules $\mathcal{F}$ . Here a map $\alpha : \widetilde M \to \mathcal{F}$ corresponds to its effect on global sections.

Proof. By Modules, Lemma 17.10.5 we have a morphism $\mathcal{F}_ M \to \widetilde M$ corresponding to the map $M \to \Gamma (X, \widetilde M) = M$ . Let $x \in X$ correspond to the prime $\mathfrak p \subset R$ . The induced map on stalks are the maps $\mathcal{O}_{X, x} \otimes _ R M \to M_{\mathfrak p}$ which are isomorphisms because $R_{\mathfrak p} \otimes _ R M = M_{\mathfrak p}$ . Hence the map $\mathcal{F}_ M \to \widetilde M$ is an isomorphism. The mapping property follows from the mapping property of the sheaves $\mathcal{F}_ M$ . $\square$

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