Definition 15.28.1. Let $R$ be a ring. Let $\varphi : E \to R$ be an $R$-module map. The Koszul complex $K_\bullet (\varphi )$ associated to $\varphi$ is the commutative differential graded algebra defined as follows:

1. the underlying graded algebra is the exterior algebra $K_\bullet (\varphi ) = \wedge (E)$,

2. the differential $d : K_\bullet (\varphi ) \to K_\bullet (\varphi )$ is the unique derivation such that $d(e) = \varphi (e)$ for all $e \in E = K_1(\varphi )$.

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