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The Stacks project

Definition 17.24.1. 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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