↑ arvau

Wikipedia / 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). It turned out to be a useful general construction in homological algebra. 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. Moreover, in certain circumstances, the complex is the complex of syzygies, that is, it tells you the relations between generators of a module, the relations between these relations, and so forth.

Definition

Let A be a commutative ring and s: Ar → A an A-linear map. Its Koszul complex Ks is

where the maps send

means the wedge product with any A-module.

Motivating example

Let M be a manifold, variety, scheme, ..., and A.

corresponds to picking r 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

. It is the tensor product of the r, so its dimensions are given by binomial coefficients.

, how do we define the locus where they all vanish? . In derived algebraic geometry, the dg intersect transversely, these are equivalent. Thus: Koszul complexes are derived intersections of zero loci.

Properties

Algebra structure

First, the Koszul complex Ks of (A,s) is a chain complex: the composition of any two maps is zero. Second, the map

makes it into a dg algebra.

As a tensor product

, then

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

Vanishing in regular case

form a regular sequence is a quasi-isomorphism, i.e.

and as for any s.

Detailed Definition

Let R be a commutative ring and E a free module of finite rank r over R for the i-th exterior power of E. Then, given an R-linear map, the Koszul complex associated to s is the chain complex of R-modules:

  1. ,

in E,

  1. .

, use the self-duality of a Koszul complex.

; this isomorphism is not canonical (for example, a choice of a volume form in differential geometry provides an example of such an isomorphism).

(i.e., an ordered basis is chosen), then, giving an R of elements in R

If M is a finitely generated R-module, then one sets:

  1. ,

.

The i-th homology of the Koszul complex

is called the i-th Koszul homology is a row vector with entries in R is

and so

Similarly,

Koszul complexes in low dimensions

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 construction, the homologies are

the annihilator of x in M. Thus, the Koszul complex and its homology encode fundamental properties of the multiplication by x is called the Koszul complex of R with respect to x, as in #Definition.

is

given by

  1. and

is applied on the right. The cycless in degree 1 are then exactly the linear relations on the elements x and 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.

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

Example

If k are indeterminates and R is the polynomial ring's forms a concrete free R-resolution of k.

Properties of a Koszul homology

Let E be a finite-rank free module over R be an R-linear map, and let t be an element of R.

, there is the exact sequence of complexes:

  1. ,

. One notes:,

In the language of homological algebra is the mapping cone.

Taking the long exact sequence of homologies, we obtain:

Here, the connecting homomorphism

where y that maps to x is a direct sum, we can simply take y to be (0, x.

The above exact sequence can be used to prove the following.

Proof by induction on r. If . Next, assume the assertion is true for r

Proof: By the theorem applied to S and S as an S is an S. So, by definition, the i by the definition of the S-module structure on M

Proof: Let S = R[y1, ..., yn]. Turn M into an S-module through the ring homomorphism S → R, yi → xi and R an S-module through and then

For a local ring, the converse of the theorem holds. More generally,

Proof: We only need to show 2. implies 1., the rest of the cycle of implications being clear. We argue by induction on r. The case r = 1 is already known. Let x denote x1, ..., xr-1. Consider

. By Nakayama's lemma and so x

Tensor products of Koszul complexes

In general, if C, D is the chain complex given by

with the differential: for any homogeneous elements x, y,

where |x| is the degree of x.

This construction applies in particular to Koszul complexes. Let E, F be two R. Then, as complexes,

To see this, it is more convenient to work with an exterior algebra

by requiring: for any homogeneous elements x, y in ΛE,

on homogeneous elements agrees with the differentials in #Definition.

as graded R-modules. Also, by the definition of a tensor product mentioned in the beginning,

Note, in particular,

  1. .

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

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. Recall that an M-regular sequence x1, ..., xn in an ideal I is maximal if I.

The Koszul homology gives a very useful characterization of a depth.

Proof: To lighten the notations, we write H(-) for H(K(-)). Let y1, ..., ys be a maximal M-regular sequence in the ideal I is clear from #Properties of a Koszul homology. From the long exact sequence of Koszul homologies and the inductive hypothesis,

  1. ,

. This completes the proof of the claim.

for all i > n - s. To conclude n - s = m, it remains to show that it is nonzero if i = n - s. is a maximal M-regular sequence in I, the ideal I, the finite union of the associated primes of the module. Thus, by prime avoidance, there is some nonzero v, which is to say,

Self-duality

There is an approach to a Koszul complex that uses a cochain complex instead of a chain complex. As it turns out, this results essentially in the same complex (the fact known as the self-duality of a Koszul complex).

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:

; that is,

is a cochain complex of free modules. This complex, also called a Koszul complex, is a complex used in . Taking the dual, there is the complex:

  1. .

coincides with the Koszul complex in the definition.

Use

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

See also

Notes

References