Theorem 15.89.18 (05ES)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.89: Formal glueing of module categories
Theorem 15.89.18 (cite)

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Theorem 15.89.18. Let $R$ be a ring, and let $f \in R$. Let $\varphi : R \to S$ be a flat ring map inducing an isomorphism $R/fR \to S/fS$. Then the functor

\[ \text{Mod}_ R \longrightarrow \text{Mod}_ S \times _{\text{Mod}_{S_ f}} \text{Mod}_{R_ f}, \quad M \longmapsto (M \otimes _ R S, M_ f, \text{can}) \]

is an equivalence.

Proof. The category appearing on the right side of the arrow is the category of triples $(M', M_1, \alpha _1)$ where $M'$ is an $S$-module, $M_1$ is a $R_ f$-module, and $\alpha _1 : M'_ f \to M_1 \otimes _ R S$ is a $S_ f$-isomorphism, see Categories, Example 4.31.3. Hence this theorem is a special case of Proposition 15.89.16. $\square$

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