Hochschild homology

Summary

Hochschild homology is a homology theory for associative algebras, with coefficients in a bimodule. It is computed from the Hochschild complex, whose $n$-th term is $M \otimes A^{\otimes n}$. In degree $0$ it gives $A/[A, A]$, the quotient of the algebra by its commutators. For smooth commutative algebras in characteristic $0$ it recovers Kähler differential forms, so it is a noncommutative analogue of differential forms.