17.24 Koszul complexes
We suggest first reading the section on Koszul complexes in More on Algebra, Section 15.28. We define the Koszul complex in the category of \mathcal{O}_ X-modules as follows.
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:
the underlying graded algebra is the exterior algebra K_\bullet (\varphi ) = \wedge (\mathcal{E}),
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 ).
Explicitly, if e_1 \wedge \ldots \wedge e_ n is a wedge product of local sections of \mathcal{E}, then
d(e_1 \wedge \ldots \wedge e_ n) = \sum \nolimits _{i = 1, \ldots , n} (-1)^{i + 1} \varphi (e_ i)e_1 \wedge \ldots \wedge \widehat{e_ i} \wedge \ldots \wedge e_ n.
It is straightforward to see that this gives a well defined derivation on the tensor algebra, which annihilates e \wedge e and hence factors through the exterior algebra.
Definition 17.24.2. Let X be a ringed space and let f_1, \ldots , f_ n \in \Gamma (X, \mathcal{O}_ X). The Koszul complex on f_1, \ldots , f_ r is the Koszul complex associated to the map (f_1, \ldots , f_ n) : \mathcal{O}_ X^{\oplus n} \to \mathcal{O}_ X. Notation K_\bullet (\mathcal{O}_ X, f_1, \ldots , f_ n), or K_\bullet (\mathcal{O}_ X, f_\bullet ).
Of course, given an \mathcal{O}_ X-module map \varphi : \mathcal{E} \to \mathcal{O}_ X, if \mathcal{E} is finite locally free, then K_\bullet (\varphi ) is locally on X isomorphic to a Koszul complex K_\bullet (\mathcal{O}_ X, f_1, \ldots , f_ n).
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