Hochschild Homology - Definition of Hochschild Homology of Algebras - Hochschild Complex

Hochschild Complex

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

Here ai is in A for all 1 ≤ i ≤ n and m ∈ M. If we let

then b ° 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.

Read more about this topic:  Hochschild Homology, Definition of Hochschild Homology of Algebras

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