Koszul Complex Of A Sequence With Coefficients

Let $R$ be a commutative unital ring, let $\mathbf{x}=(x_1,\dots ,x_n)$ be a finite ordered sequence in $R$, and let $M$ be an $R$-module. On $\bigwedge R^n$, with standard basis $e_1,\dots ,e_n$, let $\delta$ be the $R$-linear graded derivation of degree $-1$ with $\delta (e_i)=x_i$ and $\delta (R)=0$; thus $\delta (ab)=\delta (a)b+(-1{)}^{p}a\delta (b)$ for homogeneous $a$ of degree $p$.

The Koszul complex with coefficients in $M$ is $K(\mathbf{x};M)=(\bigwedge R^n{\otimes }_{R}M,d)$, where $d=\delta \otimes {\mathrm{id}}_M$. Its degree-$p$ term is ${\bigwedge }^p{R}^n{\otimes }_{R}M$ for $0\le p\le n$, and zero otherwise. Explicitly,

$d(e_{i_1}\wedge \cdots \wedge e_{i_p}\otimes m)={\sum }_{j=1}^p(-1{)}^{j-1}{e}_{i_1}\wedge \cdots \widehat{e_{i_j}}\cdots \wedge e_{i_p}\otimes x_{i_j}m\mathrm{.}$

The empty wedge is $1\in R$. In particular $d(e_i\otimes m)=1\otimes x_{i}m$ in $K_0=R{\otimes }_{R}M$, canonically identified with $M$ by $r\otimes m\mapsto rm$ with inverse $m\mapsto 1\otimes m$. The differential out of degree zero is zero. The derivation respects the exterior relations and the two orders of deleting distinct factors cancel in $d^2$, so these maps form a chain complex. For $n=0$ this is $M$ in degree zero under the same identification.