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. Mathematics is the study of quantity, structure, space and change. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...

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: In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... In abstract algebra, a module is a generalization of a vector space. ... In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. ...

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, H₁(K_•(x)) = Ann_R(x). Annihilators are a concept that occurs in ring theory, a branch of mathematics. ...

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. In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. ... The word category (plural categories; from Greek κατηγορια meaning assertion or accusation, hence categorical denial) has several meanings: it is used informally to mean a class of things, as in the category of all living things. See categorization. ...

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. In commutative algebra, if R is a commutative ring and M an R-module, a R-regular sequence on M is a d-tuple of (non-zero non-unit) elements r1, r2, ..., rd from R such that for each i, ri is not a zerodivisor on the quotient R-module...

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.