Definitions

# 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:


0to Rxrightarrow{ x }Rto0.

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 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 jth degree in the complex (0 ≤ jn). The matrices involved in the maps can be written down precisely. Letting $e_\left\{i_1...i_p\right\}$ denote a free-basis generator in Kp, d: Kp $mapsto$ Kp − 1 is defined by:


d(e_{i_1...i_p}) := sum _{j=1}^{p}(-1)^{j-1}x_{i_j}e_{i_1...widehat{i_j}...i_p}.

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


0 to R xrightarrow{ d_2 } R^2 xrightarrow{ d_1 } Rto 0, with the matrices $d_1$ and $d_2$ given by


d_1 = begin{bmatrix} x & y end{bmatrix} and

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, so K(x1, x2, ..., xn) forms a free resolution of the R-module R/(x1, x2, ..., xn)R.

## 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

If (R, m) is a local ring and M is a finitely-generated R-module with x1, x2, ..., xn in m, then the following are equivalent:

1. The (xi) form a regular sequence on M,
2. H1(K(xi)) = 0,
3. Hj(K(xi)) = 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.

## 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

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