(Co-)homology of Rees semigroup algebras

TL;DR

This paper demonstrates that the Hochschild homology and cohomology of Rees semigroup algebras are isomorphic to those of the associated group algebra, using Morita equivalence.

Contribution

It establishes a Morita equivalence-based isomorphism between the homology and cohomology of Rees semigroup algebras and their underlying group algebras.

Findings

Hochschild homology of $\, ext{l}^1(S)$ matches that of the group algebra.

Hochschild cohomology of $\, ext{l}^1(S)$ matches that of the group algebra.

Morita equivalence links the algebraic properties of Rees semigroup algebras to group algebras.

Abstract

Let $S$ be a Rees semigroup, and let $\ell^1(S)$ be its convolution semigroup algebra. Using Morita equivalence we show that bounded Hochschild homology and cohomology of $\ell^1(S)$ is isomorphic to those of the underlying discrete group algebra.