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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