Lemma 15.89.15 (05EQ)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.89: Formal glueing of module categories
Lemma 15.89.15 (cite)

Lemma 15.89.15. Let $\varphi : R \to S$ be a flat ring map and $I = (f_1, \ldots , f_ t)$ and ideal. Let $R \to R'$ be a flat ring map, and set $S' = S \otimes _ R R'$. Then we obtain a commutative diagram of categories and functors

\[ \xymatrix{ \text{Mod}_ R \ar[r]_-{\text{Can}} \ar[d]_{-\otimes _ R R'} & \text{Glue}(R \to S, f_1, \ldots , f_ t) \ar[r]_-{H^0} \ar[d]^{-\otimes _ R R'} & \text{Mod}_ R \ar[d]^{-\otimes _ R R'} \\ \text{Mod}_{R'} \ar[r]^-{\text{Can}} & \text{Glue}(R' \to S', f_1, \ldots , f_ t) \ar[r]^-{H^0} & \text{Mod}_{R'} } \]

Proof. Omitted. $\square$

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