The Stacks project

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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