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:

$0\to R\xrightarrow{\ x\ }R\to0.$

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)) = Ann_R(x).

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

Now, if x₁, x₂, ..., x_n are elements of R, the Koszul complex of R with respect to x₁, x₂, ..., x_n, usually denoted K_•(x₁, x₂, ..., x_n), is the tensor product (in the category of R-complexes) $K_\bullet(x_1) \otimes K_\bullet(x_2) \otimes \cdots \otimes K_\bullet(x_n)$ 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 $K_p$ is a free R-module of rank $\dbinom{n}{p}$ (thus, it is zero unless 0 ≤ p ≤ n); this module has a basis $\left(e_{i_1,...,i_p}\right)_{1 \leq i_1 < i_2 < \cdots < i_p \leq n}$. The element $e_{i_1,...,i_p}$ is defined as the pure tensor $f_1 \otimes f_2 \otimes \cdots \otimes f_n \in K_\bullet(x_1) \otimes K_\bullet(x_2) \otimes \cdots \otimes K_\bullet(x_n)$, where for every 1 ≤ j ≤ n, we let $f_j$ be the generator 1 of $K_1(x_j)$ if $j \in \{i_1, ..., i_p\}$ and the generator 1 of $K_0(x_j)$ otherwise.

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

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

where $i_1,...,\widehat{i_j},...,i_p$ means $i_1,...,i_{j-1},i_{j+1},...,i_p$ (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

$0 \to R \xrightarrow{\ d_2\ } R^2 \xrightarrow{\ d_1\ } R\to 0,$

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 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 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, ..., x_n form a regular sequence, the higher homology modules of the Koszul complex are all zero.

Example

If k is a field and X₁, X₂, ..., X_d are indeterminates and R is the polynomial ring k[X₁, X₂, ..., 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 x₁, x₂, ..., x_n are elements of the maximal ideal m, then the following are equivalent:

1. The (x_i) form a regular sequence on M,
2. 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 commuting bounded linear operators in a Banach space.

See also

• Koszul–Tate complex

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