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).
Definition of Hochschild homology of algebras Let k be a field, A an associative k-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product A e = A ⊗ A o {\displaystyle A^{e}=A\otimes A^{o}} 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
H H n ( A , M ) = Tor n A e ( A , M ) {\displaystyle HH_{n}(A,M)=\operatorname {Tor} _{n}^{A^{e}}(A,M)}
H H n ( A , M ) = Ext A e n ( A , M ) {\displaystyle HH^{n}(A,M)=\operatorname {Ext} _{A^{e}}^{n}(A,M)}
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 {\displaystyle A^{\otimes n}} for the n-fold tensor product of A over k. The chain complex that gives rise to Hochschild homology is given by
C n ( A , M ) := M ⊗ A ⊗ n {\displaystyle C_{n}(A,M):=M\otimes A^{\otimes n}}
with boundary operator d i {\displaystyle d_{i}} defined by
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