Lemma 15.7.5 (07RW)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.7: Fibre products of rings, III
Lemma 15.7.5 (cite)

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Lemma 15.7.5. With $A, A', B, B', C, C', D, D', I$ as in Situation 15.7.1.

Let $(N, M', \varphi )$ be an object of $\text{Mod}_ D \times _{\text{Mod}_ C} \text{Mod}_{C'}$. If $M'$ is flat over $A'$ and $N$ is flat over $B$, then $N' = N \times _{\varphi , M} M'$ is flat over $B'$.
If $L'$ is a $D'$-module flat over $B'$, then $L' = (L \otimes _{D'} D) \times _{(L \otimes _{D'} C)} (L \otimes _{D'} C')$.
The category of $D'$-modules flat over $B'$ is equivalent to the categories of objects $(N, M', \varphi )$ of $\text{Mod}_ D \times _{\text{Mod}_ C} \text{Mod}_{C'}$ with $N$ flat over $B$ and $M'$ flat over $A'$.

Proof. Part (1) follows from part (1) of Lemma 15.6.8.

Part (2) follows from part (2) of Lemma 15.6.8 using that $L' \otimes _{D'} D = L' \otimes _{B'} B$, $L' \otimes _{D'} C' = L' \otimes _{B'} A'$, and $L' \otimes _{D'} C = L' \otimes _{B'} A$, see discussion in Situation 15.7.1.

Part (3) is an immediate consequence of (1) and (2). $\square$

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