Koszul complex

In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology).

As a tool, its homology can be used to tell when a set of elements of a (local) ring is an M-regular sequence, and hence it can be used to prove basic facts about the depth of a module or ideal which is an algebraic notion of dimension that is related to but different from the geometric notion of Krull dimension.

Let M be a manifold, variety, scheme, ..., and A be the ring of functions on it, denoted

When r = 1, the Koszul complex is whose cokernel is the ring of functions on the zero locus f = 0.

In general, the Koszul complex is The cokernel of the last map is again functions on the zero locus

In algebraic geometry, the ring of functions of the zero locus is

In derived algebraic geometry, the dg ring of functions is the Koszul complex.

Thus: Koszul complexes are derived intersections of zero loci.

denotes the derived tensor product of chain complexes of A-modules.

The Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology).

Let R be a commutative ring and E a free module of finite rank r over R. We write

If M is a finitely generated R-module, then one sets: which is again a chain complex with the induced differential

is and so Similarly, Given a commutative ring R, an element x in R, and an R-module M, the multiplication by x yields a homomorphism of R-modules, Considering this as a chain complex (by putting them in degree 1 and 0, and adding zeros elsewhere), it is denoted by

With more elements the higher-dimensional Koszul homologies measure the higher-level versions of this.

form a regular sequence, the higher homology modules of the Koszul complex are all zero.

Taking the long exact sequence of homologies, we obtain: Here, the connecting homomorphism is computed as follows.

More generally, Theorem — [7] Let R be a ring and M a nonzero finitely generated module over R .

are elements of the Jacobson radical of R, then the following are equivalent: Proof: We only need to show 2. implies 1., the rest being clear.

is the chain complex given by with the differential: for any homogeneous elements x, y, where |x| is the degree of x.

by requiring: for any homogeneous elements x, y in ΛE, One easily sees that

Note, in particular, The next proposition shows how the Koszul complex of elements encodes some information about sequences in the ideal generated by them.

Proposition — Let R be a ring and I = (x1, ..., xn) an ideal generated by some n-elements.

Proof: (Easy but omitted for now) As an application, we can show the depth-sensitivity of a Koszul homology.

Given a finitely generated module M over a ring R, by (one) definition, the depth of M with respect to an ideal I is the supremum of the lengths of all regular sequences of elements of I on M. It is denoted by

Recall that an M-regular sequence x1, ..., xn in an ideal I is maximal if I contains no nonzerodivisor on

From the long exact sequence of Koszul homologies and the inductive hypothesis, which is

is a maximal M-regular sequence in I, the ideal I is contained in the set of all zerodivisors on

Let E be a free module of finite rank r over a ring R. Then each element e of E gives rise to the exterior left-multiplication by e: Since

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