The Stacks project

Lemma 15.88.13. Let $\varphi : R \to S$ be a flat ring map and $(f_1, \ldots , f_ t) = R$. Then $\text{Can}$ and $H^0$ are quasi-inverse equivalences of categories

\[ \text{Mod}_ R = \text{Glue}(R \to S, f_1, \ldots , f_ t) \]

Proof. Consider an object $\mathbf{M} = (M', M_ i, \alpha _ i, \alpha _{ij})$ of $\text{Glue}(R \to S, f_1, \ldots , f_ t)$. By Algebra, Lemma 10.24.5 there exists a unique module $M$ and isomorphisms $M_{f_ i} \to M_ i$ which recover the glueing data $\alpha _{ij}$. Then both $M'$ and $M \otimes _ R S$ are $S$-modules which recover the modules $M_ i \otimes _ R S$ upon localizing at $f_ i$. Whence there is a canonical isomorphism $M \otimes _ R S \to M'$. This shows that $\mathbf{M}$ is in the essential image of $\text{Can}$. Combined with Lemma 15.88.11 the lemma follows. $\square$


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