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 $K_\bullet (\mathcal{O}_ X, f_1, \ldots , f_ n)$.
Comments (0)