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.
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:
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 denote a free-basis generator in Kp, d: Kp Template:Mapsto Kp − 1 is defined by:
For the case of two elements x and y, the Koszul complex can then be written down quite succinctly as
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.
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.
- The (xi) form a regular sequence on M,
- Hj(K•(xi)) = 0 for all j ≥ 1.