In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul . It turned out to be a useful general construction in homological algebra.

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 →^xR → 0.

Call this complex K_•(x).

Counting the right-hand copy of R as the zeroth slot and the left-hand copy as the first slot, this 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, H₀(K_•(x)) = Ann_R(x).

This complex K_•(x) is 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 complex. There are exactly (n choose j) copies of the ring R in the jth slot in the complex (0 ≤ j ≤ n). The matrices involved in the maps can be written down precisely. Letting denote a free-basis generator in is defined by:

For the case of two elements x and y, the Koszul complex can then be written down quite succinctly as 0 → R →^φR² → ^ψR →0, with the matrices φ and ψ given by and respectively. The cycles in slot 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, x_n, ..., x_n)R.

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.

Theorem

If (R,m) is local and M is a finitely-generated R-module with x₁, x₂, ...,x_n in m, then the following are equivalent:
1) The (x_i) form an M-sequence,
2) H₁(K_•(x_i)) = 0,
3) H_j(K_•(x_i)) = 0 for all j ≥ 1.