Hochschild homology is an important concept in algebraic topology and homological algebra, often used to study algebraic structures, particularly associative algebras. It was introduced by Gerhard Hochschild in the 1940s. ### Definition and Construction Hochschild homology is typically defined for a unital associative algebra \( A \) over a field \( k \) (or more generally, over a commutative ring).
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For a unital associative -algebra and an -bimodule , Hochschild homology is the homology of the complex . Its boundary multiplies neighboring algebra entries, with the first and last terms using the two actions on .
In degree zero, . In particular, is the vector-space quotient by commutators. If is commutative, this degree-zero quotient is simply . Higher groups retain information lost by that quotient and connect algebraic multiplication to differential forms in smooth commutative settings.