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.

Introduction[edit]

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:RR 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. For every p, its pth degree entry is a free R-module of rank (thus, it is zero unless 0 ≤ pn); this module has a basis . The element is defined as the pure tensor , where for every 1 ≤ jn, we let be the generator 1 of if and the generator 1 of otherwise.

The boundary map of the Koszul complex can be written explicitly with respect to this basis. Namely, the R-linear map is defined by:

where means (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

with the matrices and given by

and

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.

Example[edit]

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.

Theorem[edit]

Let (R, m) be a Noetherian local ring with maximal ideal m, and let M be a finitely-generated R-module. If x1, x2, ..., xn are elements of the maximal ideal m, then the following are equivalent:

  1. The (xi) form a regular sequence on M,
  2. Hj(K(xi)) = 0 for all j ≥ 1.

Applications[edit]

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

See also[edit]

References[edit]