Definition 17.23.1. Let $X$ be a ringed space. Let $\varphi : \mathcal{E} \to \mathcal{O}_ X$ be an $\mathcal{O}_ X$-module map. The Koszul complex $K_\bullet (\varphi )$ associated to $\varphi$ is the sheaf of commutative differential graded algebras defined as follows:

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

2. the differential $d : K_\bullet (\varphi ) \to K_\bullet (\varphi )$ is the unique derivation such that $d(e) = \varphi (e)$ for all local sections $e$ of $\mathcal{E} = K_1(\varphi )$.

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