Lemma 10.71.8 (00LV)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.71: Ext groups
Lemma 10.71.8 (cite)

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Lemma 10.71.8. Let $R$ be a ring. Let $M$ , $N$ be $R$ -modules. Any $x\in R$ such that either $xN = 0$ , or $xM = 0$ annihilates each of the modules $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N)$ .

Proof. Pick a free resolution $F_{\bullet }$ of $M$ . Since $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N)$ is defined as the cohomology of the complex $\mathop{\mathrm{Hom}}\nolimits _ R(F_{\bullet }, N)$ the lemma is clear when $xN = 0$ . If $xM = 0$ , then we see that multiplication by $x$ on $F_{\bullet }$ lifts the zero map on $M$ . Hence by Lemma 10.71.5 we see that it induces the same map on Ext groups as the zero map. $\square$

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