Lemma 15.28.5 (0626)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.28: The Koszul complex
Lemma 15.28.5 (cite)

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Lemma 15.28.5. Let $R$ be a ring. Let $\varphi : E \to R$ be an $R$-module map. Let $e \in E$ with image $f = \varphi (e)$ in $R$. Then

\[ f = de + ed \]

as endomorphisms of $K_\bullet (\varphi )$.

Proof. This is true because $d(ea) = d(e)a - ed(a) = fa - ed(a)$. $\square$

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