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The Stacks project

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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View Definition 15.28.1 as pdf