Koszul complex of a module

Definition

Let $R$ be a commutative unital ring and $N$ be a $R$-module. Let $x\in N$. Then the Koszul complex of $x$, denoted $K(x)$, is the complex of $R$-modules:

$K(x):0\to R\to N\to {\mathrm{\Lambda }}^{2}N\to {\mathrm{\Lambda }}^{3}N\to \dots$

where the general map from ${\mathrm{\Lambda }}^{i}N$ to ${\mathrm{\Lambda }}^{i+1}N$ takes as input $a$ and outputs $x\wedge a$.

Related notions

• Koszul complex of a sequence of elements