Hochschild homology

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 Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956). 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 Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956). Let k be a field, A an associative k-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product Ae=A⊗Ao 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. Cartan & Eilenberg (1956) 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 A⊗n 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 di defined by where ai is in A for all 1 ≤ i ≤ n and m ∈ M. If we let then b ∘ b = 0 , {displaystyle bcirc b=0,} so (Cn(A,M), b) is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M. The maps di are face maps making the family of modules Cn(A,M) a simplicial object in the category of k-modules, i.e. a functor Δo → k-mod, where Δ is the simplex category and k-mod is the category of k-modules. Here Δo is the opposite category of Δ. The degeneracy maps are defined by