Hochschild homology and cohomology

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The Hochschild homology , named after Gerhard Hochschild , is a mathematical theory, specifically to the study of algebra is tailored. It is a homology or cohomology theory that results from chain complexes or coquette complexes that are closely related to the algebra structure.

Construction of the homology groups

In the following we consider an associative algebra with one element over a field , in short a K -algebra . Furthermore, a be - bimodule given, that is, the module elements from the left and right are multiplied with elements of algebra, so that the corresponding left and right module structures are compatible, which for all and means. If one denotes the -fold tensor product of with itself, where , the following maps can be defined:

wherein the at K -linear pictures continue. Be further , that is

and so on. Then applies to all , that is, you get a chain complex

.

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

,

where was set. Since the above definitions make use of the algebra and bimodular structure, the Hochschild homology groups can contain information about the algebra .

Construction of the cohomology groups

The Hochschild cohomology groups are obtained by an analog construction from spaces of -linear homomorphisms , where again the considered -algebra and a -Bimodule are assumed. For one receives .

We are again defining images

.

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

One sets, this time with an upper index:

,

this means

and so on. Then applies to everyone . So you get a coquette complex

.

The Hochschild cohomology of with values ​​in is defined as the cohomology of this coquette complex, i.e. the -th Hochschild cohomology group of with values ​​in is the factor group

,

where is the null morphism .

Here, too, the algebra structure of goes into the definitions, so that the Hochschild cohomology groups contain information about the algebra.

Examples

In the following examples, which are intended to prove information contained in the Hochschild homomology and cohomology groups, let us again assume an associative algebra with a one element and a bimodule. The 0th Hochschild homology and cohomology groups can be easily determined:

,

where , the commutator of and , is the product of all .

Next is

.

is a -Bimodule in a natural way , whereby the compatibility condition is given by the associative law. One therefore receives as a special case

and ,

where is the center of .

A - derivation on with values ​​in is a -linear mapping with the additional property that reminds of the product rule for the derivation. With is the set of all derivations referred. For each is by given such derivation. Such derivatives are called internal derivatives , denote the set of all internal derivatives. An inspection of the formulas given above for and shows

,

and therefore

.

The first Hochschild cohomology group provides information about the richness of the derivatives, their disappearance means that all derivatives are internal.

Multilinear maps

The Hochschild cohomology groups can alternatively be introduced using the spaces of the multilinear mappings . One sets for and :

and one arrives at a corresponding coquette complex

,

with which one can define cohomology groups again. One obtains isomorphic groups for the above defined , since multilinear maps and linear maps correspond 1 to 1 after construction of the tensor product.

Topological algebras

The concepts presented above can also be carried out for topological algebras , in particular Banach algebras , whereby the projective tensor product is used for the formation of tensor products in the case of Banach algebras , and all mappings are limited to continuous maps.

literature

  • Henri Cartan , Samuel Eilenberg : Homological Algebra , Princeton University Press (1999), ISBN 978-0-691-04991-5 , especially Chapter X
  • AY Helemskii: The Homology of Banach and Topological Algebras. Kluwer Academic Publishers (1989), ISBN 0-7923-0217-6 .
  • G. Hochschild: On the Cohomology Groups of an Associative Algebra , Annals of Mathematics Volume 46 (1945), pages 58-76