 Hochschild homology

For a unital associative $k$-algebra $A$ and an $A$-bimodule $M$, Hochschild homology is the homology of the complex $C_n(A,M)=M{\otimes }_k{A}^{\otimes n}$. Its boundary multiplies neighboring algebra entries, with the first and last terms using the two actions on $M$.

In degree zero, $H{H}_0(A,M)=M/⟨am-ma:a\in A,m\in M{⟩}_k$. In particular, $H{H}_0(A,A)$ is the vector-space quotient by commutators. If $A$ is commutative, this degree-zero quotient is simply $A$. Higher groups retain information lost by that quotient and connect algebraic multiplication to differential forms in smooth commutative settings.

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).