Koszul complex

Definition

Let $R$ be a commutative ring and let $x = (x_1, \ldots, x_n)$ be a sequence of elements of $R$. Let $e_1, \ldots, e_n$ be the standard basis of the free module $R^n$. For $0 \le p \le n$, let $K_p = \Lambda^p R^n$ be its $p$-th exterior power. It is free with basis the elements $e_{i_1} \wedge \cdots \wedge e_{i_p}$ for $i_1 < \cdots < i_p$. The Koszul complex $K_\bullet(x)$ is $$0 \to K_n \xrightarrow{d_n} K_{n-1} \to \cdots \to K_1 \xrightarrow{d_1} K_0 \to 0,$$ with $$d_p(e_{i_1} \wedge \cdots \wedge e_{i_p}) = \sum_{j=1}^{p} (-1)^{j+1} x_{i_j}\, e_{i_1} \wedge \cdots \wedge \widehat{e_{i_j}} \wedge \cdots \wedge e_{i_p},$$ where the hat marks an omitted factor. For an $R$-module $M$, the Koszul complex of $x$ on $M$ is $K_\bullet(x; M) = K_\bullet(x) \otimes_R M$.