Lemma 10.73.1 (087N)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.73: Functorialities for Ext
Lemma 10.73.1 (cite)

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Lemma 10.73.1. Given a flat ring map $R \to R'$ , an $R$ -module $M$ , and an $R'$ -module $N'$ the natural map

$\mathop{\mathrm{Ext}}\nolimits ^ i_{R'}(M \otimes _ R R', N') \to \text{Ext}^ i_ R(M, N')$

is an isomorphism for $i \geq 0$ .

Proof. Choose a free resolution $F_\bullet$ of $M$ . Since $R \to R'$ is flat we see that $F_\bullet \otimes _ R R'$ is a free resolution of $M \otimes _ R R'$ over $R'$ . The statement is that the map

$\mathop{\mathrm{Hom}}\nolimits _{R'}(F_\bullet \otimes _ R R', N') \to \mathop{\mathrm{Hom}}\nolimits _ R(F_\bullet , N')$

induces an isomorphism on homology groups, which is true because it is an isomorphism of complexes by Lemma 10.14.3. $\square$

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