Definition of Hochschild Homology of Algebras
Let k be a ring, 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
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