Koszul complex

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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.

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[edit] 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 from R to itself, usually denoted R →x R. It is useful to throw in zeroes on each end and make this a (free) R-complex:

0 → R →x R → 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 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 ≤ j ≤ n). The matrices involved in the maps can be written down precisely. Letting e_{i_1...i_n} denote a free-basis generator in Kp, d: Kp \mapsto Kp − 1 is defined by:

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

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

0 \to R \rightarrow^{d_2} R^2 \to^{d_1} R\to 0,

with the matrices d1 and d2 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.

[edit] 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 on the Xi's K•(Xi) forms a concrete free R-resolution of k.

[edit] 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.

[edit] Applications

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

[edit] References

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