The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

17.21 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 17.21.1. 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 $\mathcal{E} = K_1(\varphi )$.

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 17.21.2. 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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