# 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*:*R* →*R* from *R* to itself. It is useful to throw in zeroes on each end and make this a (free) *R*-complex:

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*, H_{0}(*K*_{•}(*x*)) = *R*/*xR*, and its first homology is exactly the annihilator of *x*, H_{1}(*K*_{•}(*x*)) = Ann_{R}(*x*).

This chain complex *K*_{•}(*x*) is called the **Koszul complex** of *R* with respect to *x*.

Now, if *x*_{1}, *x*_{2}, ..., *x*_{n} are elements of *R*, the **Koszul complex** of *R* with respect to *x*_{1}, *x*_{2}, ..., *x*_{n}, usually denoted *K*_{•}(*x*_{1}, *x*_{2}, ..., *x*_{n}), is the tensor product in the category of *R*-complexes of the Koszul complexes defined above individually for each *i*.

The Koszul complex is a free chain complex. There are exactly (*n* choose *j*) copies of the ring *R* in the *j*th degree in the complex (0 ≤ *j* ≤ *n*). The matrices involved in the maps can be written down precisely. Letting denote a free-basis generator in
*K*_{p}, *d*: *K*_{p} Template:Mapsto *K*_{p − 1} is defined by:

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

with the matrices and given by

Note that *d _{i}* 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 H

_{1}(

*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 *x*_{1}, *x*_{2}, ..., *x*_{n} form a regular sequence, the higher homology modules of the Koszul complex are all zero.

## Example

If *k* is a field and *X*_{1}, *X*_{2}, ..., *X*_{d} are indeterminates and *R* is the polynomial ring *k*[*X*_{1}, *X*_{2}, ..., *X*_{d}], the Koszul complex *K*_{•}(*X*_{i}) on the *X*_{i}'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 *x*_{1}, *x*_{2}, ..., *x*_{n} are elements of the maximal ideal *m*, then the following are equivalent:

- The (
*x*_{i}) form a regular sequence on*M*, - H
_{j}(*K*_{•}(*x*_{i})) = 0 for all*j*≥ 1.

## Applications

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

## See also

## References

- David Eisenbud,
*Commutative Algebra. With a view toward algebraic geometry*, Graduate Texts in Mathematics, vol 150, Springer-Verlag, New York, 1995. ISBN 0-387-94268-8