Hochschild homology

The Hochschild homology and cohomology, named after Gerhard Hochschild, is a mathematical theory that is specifically tailored to the study of algebras. There is a homology or cohomology theory, which arises from chain complexes or Kokettenkomplexen, are closely related with the algebra structure.

Construction of the homology groups

We consider in the following an associative algebra with identity over a field, just a K- algebra. Furthermore, a - bimodule is given, that is, the module elements can be multiplied from the left and right with elements of the algebra, so that the associated left and right module structures are compatible, which means and for all. If we denote by the -fold tensor product of with itself, which, as the following figures can be defined:

Wherein the resume to K- linear maps. Further is, that is,

And so on. Then for all, ie one obtains a chain complex

The Hochschild homology of with values ​​in is defined as the homology of this chain complex, ie the -th Hochschild homology group of values ​​in the factor group

Was being set. As the above definitions make the of the algebras and Bimodulstruktur use, the Hochschild homology groups can contain information about the algebra.

Construction of the cohomology groups

The Hochschild - cohomology groups are obtained by an analogous construction of spaces of linear homomorphisms, which again were the considered algebra and a - bimodule. For obtained.

We define again pictures

If so we need to decide how to act and thereby results in an item, and it goes like this

Man sets, this time with an upper index:

That is

And so on. Then for all. One thus gets a Kokettenkomplex

The Hochschild cohomology of with values ​​in is defined as the cohomology of this Kokettenkomplexex, ie the -th Hochschild cohomology group of with - values ​​in the factor group

Where the Nullmorphismus is.

Again, the algebra structure is one of the definitions so that the Hochschild cohomology groups - contains information about the algebra.

Examples

In the following examples, which are situated in the Hochschild cohomology groups Homomlogie and plug the newsletter, and be again an associative algebra with unit element and a - bimodule. The 0-th Hochschild homology and cohomology can easily be determined:

Wherein the commutator, and that is the product of all is.

Next is

Is naturally a - bimodule, where the compatibility condition is given exactly by the associative law. As a special case we obtain therefore

Where the center is.

A - Derivation on with values ​​in is a linear map with the additional property that is reminiscent of the product rule for the derivation. With the set of all derivations is denoted. For each is given by such a derivation. Such derivations are called inner derivations, denote the set of all inner derivations. Shows an inspection of the above formulas for and

And therefore

The first Hochschild cohomology group - there is information about the richness of the derivations, their disappearance means that all derivations are inner.

Multi- linear maps

The Hochschild - cohomology groups can be introduced as an alternative by means of the spaces of multilinear mappings. It is for and:

And you come to a corresponding Kokettenkomplex

With which one can define cohomology groups again. Obtained for the above-defined isomorphic groups, as multi- linear maps and linear maps is correct to the construction of the tensor product 1 to 1.

Topological algebras

The above presented concepts can be also for topological algebras, in particular Banach algebras, execute, where one uses the projective tensor product in the Tensorproduktbildung in the case of Banach algebras and is limited in all pictures appearing on continuous maps.