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.

1 Introduction
2 Example
3 Theorem
4 Applications
5 References

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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)) = 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 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 <math>e_{i_1...i_n}</math> denote a free-basis generator in K_p, d: K_p <math>\mapsto</math> K_{p − 1} is defined by:

<math>

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

For the case of two elements x and y, the Koszul complex can then be written down quite succinctly as 0 → R →^φ R² →^ψ R → 0, with the matrices ψ and φ given by

<math>\begin{bmatrix}

-y & x\\ \end{bmatrix} </math> and

<math>\begin{bmatrix}

x\\ y\\ \end{bmatrix}, </math>

respectively. 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, so K_•(x₁, x₂, ..., x_n) forms a free resolution of the R-module R/(x₁, x₂, ..., x_n)R.

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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 on the X_i's K_•(X_i) forms a concrete free R-resolution of k.