Koszul complex

Summary

The Koszul complex of elements $x_1, \ldots, x_n$ of a commutative ring $R$ is a complex of free modules built from the exterior algebra of $R^n$. Its zeroth homology is $R/(x_1, \ldots, x_n)$, and when the $x_i$ form a regular sequence it is a free resolution of this quotient. It computes depth, gives the proof of Hilbert's syzygy theorem, and is a basic tool throughout commutative algebra.