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Concept
Hochschild homology
Summary
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, and extended to algebras over more general rings by .
Let k be a field, A an associative k-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product of A with its opposite algebra. Bimodules over A are essentially the same as modules over the enveloping algebra of A, so in particular A and M can be considered as Ae-modules. defined the Hochschild homology and cohomology group of A with coefficients in M in terms of the Tor functor and Ext functor by
Let k be a ring, A an associative k-algebra that is a projective k-module, and M an A-bimodule. We will write for the n-fold tensor product of A over k. The chain complex that gives rise to Hochschild homology is given by
with boundary operator defined by
where is in A for all and . If we let
then , so is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M.
The maps are face maps making the family of modules a simplicial object in the of k-modules, i.e., a functor Δo → k-mod, where Δ is the and k-mod is the category of k-modules. Here Δo is the of Δ. The degeneracy maps are defined by
Hochschild homology is the homology of this simplicial module.
There is a similar looking complex called the Bar complex which formally looks very similar to the Hochschild complexpg 4-5. In fact, the Hochschild complex can be recovered from the Bar complex asgiving an explicit isomorphism.
There's another useful interpretation of the Hochschild complex in the case of commutative rings, and more generally, for sheaves of commutative rings: it is constructed from the derived self-intersection of a scheme (or even derived scheme) over some base scheme . For example, we can form the derived fiber productwhich has the sheaf of derived rings .
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