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. 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)) = AnnR(x).
This complex K•(x) is 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 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 <math>e_{i_1...i_n}<math> denote a free-basis generator in <math>K_p, d:K_p \to K_{p-1}<math> is defined by:
<math> d(e_{i_1...i_n}) := \sum _{j=1}^{p}(-1)^{j-1}x_{i_j}e_{i_1...\hat{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 →φR2 → ψR →0, with the matrices φ and ψ given by <math>\begin{bmatrix} -y & x\\ \end{bmatrix} <math> and <math>\begin{bmatrix} x\\ y\\ \end{bmatrix} <math> 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, ..., 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, xn, ..., xn)R.
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.
Theorem
If (R,m) is local and M is a finitely-generated R-module with x1, x2, ...,xn in m, then the following are equivalent:
1) The (xi) form an M-sequence,
2) H1(K•(xi)) = 0,
3) Hj(K•(xi)) = 0 for all j ≥ 1.