Hochschild chain complex

For a $k$-algebra $R$ and an $R$-$R$-bimodule $M$, the Hochschild chains areTheir boundary isAssociativity and the bimodule axioms give $b^2=0$.

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.