# Koszul complex

In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra.

## Introduction

In commutative algebra, if x is an element of the ring R, multiplication by x is R-linear and so represents an R-module homomorphism x:RR from R to itself. It is useful to throw in zeroes on each end and make this a (free) R-complex:

${\displaystyle 0\to R{\xrightarrow {\ x\ }}R\to 0.}$

Call this chain complex K(x).

Counting the right-hand copy of R as the zeroth degree and the left-hand copy as the first degree, this chain complex neatly captures the most important facts about multiplication by x because its zeroth homology is exactly the homomorphic image of R modulo the multiples of x, H0(K(x)) = R/xR, and its first homology is exactly the annihilator of x, H1(K(x)) = AnnR(x).

This chain complex K(x) is called the Koszul complex of R with respect to x.

Now, if x1, x2, ..., xn are elements of R, the Koszul complex of R with respect to x1, x2, ..., xn, usually denoted K(x1, x2, ..., xn), is the tensor product (in the category of R-complexes) ${\displaystyle K_{\bullet }(x_{1})\otimes K_{\bullet }(x_{2})\otimes \cdots \otimes K_{\bullet }(x_{n})}$ of the Koszul complexes defined above individually for each i.

The Koszul complex is a free chain complex. For every p, its pth degree entry ${\displaystyle K_{p}}$ is a free R-module of rank ${\displaystyle {\dbinom {n}{p}}}$ (thus, it is zero unless 0 ≤ pn); this module has a basis ${\displaystyle \left(e_{i_{1},...,i_{p}}\right)_{1\leq i_{1}. The element ${\displaystyle e_{i_{1},...,i_{p}}}$ is defined as the pure tensor ${\displaystyle f_{1}\otimes f_{2}\otimes \cdots \otimes f_{n}\in K_{\bullet }(x_{1})\otimes K_{\bullet }(x_{2})\otimes \cdots \otimes K_{\bullet }(x_{n})}$, where for every 1 ≤ jn, we let ${\displaystyle f_{j}}$ be the generator 1 of ${\displaystyle K_{1}(x_{j})}$ if ${\displaystyle j\in \{i_{1},...,i_{p}\}}$ and the generator 1 of ${\displaystyle K_{0}(x_{j})}$ otherwise.

The boundary map of the Koszul complex can be written explicitly with respect to this basis. Namely, the R-linear map ${\displaystyle d:K_{p}\to K_{p-1}}$ is defined by:

${\displaystyle d(e_{i_{1},...,i_{p}}):=\sum _{j=1}^{p}(-1)^{j-1}x_{i_{j}}e_{i_{1},...,{\widehat {i_{j}}},...,i_{p}},}$

where ${\displaystyle i_{1},...,{\widehat {i_{j}}},...,i_{p}}$ means ${\displaystyle i_{1},...,i_{j-1},i_{j+1},...,i_{p}}$ (that is, the j-th term is being omitted).

For the case of two elements x and y, the Koszul complex can then be written down quite succinctly as

${\displaystyle 0\to R{\xrightarrow {\ d_{2}\ }}R^{2}{\xrightarrow {\ d_{1}\ }}R\to 0,}$

with the matrices ${\displaystyle d_{1}}$ and ${\displaystyle d_{2}}$ given by

${\displaystyle d_{1}={\begin{bmatrix}x&y\\\end{bmatrix}}}$ and
${\displaystyle d_{2}={\begin{bmatrix}-y\\x\\\end{bmatrix}}.}$

Note that di is applied on the left. The cycles in degree 1 are then exactly the linear relations on the elements x and y, while the boundaries are the trivial relations. The first Koszul homology H1(K(x, y)) therefore measures exactly the relations mod the trivial relations. With more elements the higher-dimensional Koszul homologies measure the higher-level versions of this.

In the case that the elements x1, x2, ..., xn form a regular sequence, the higher homology modules of the Koszul complex are all zero.

## Example

If k is a field and X1, X2, ..., Xd are indeterminates and R is the polynomial ring k[X1, X2, ..., Xd], the Koszul complex K(Xi) on the Xi's forms a concrete free R-resolution of k.

## Theorem

Let (R, m) be a Noetherian local ring with maximal ideal m, and let M be a finitely-generated R-module. If x1, x2, ..., xn are elements of the maximal ideal m, then the following are equivalent:

1. The (xi) form a regular sequence on M,
2. Hj(K(xi)) = 0 for all j ≥ 1.

## Applications

The Koszul complex is essential in defining the joint spectrum of a tuple of commuting bounded linear operators in a Banach space.