The Hochschild cohomology ring of a selfinjective algebra of finite representation type

This paper describes the Hochschild cohomology ring of a selfinjective algebra $\Lambda$ of finite representation type over an algebraically closed field $K$, showing that the quotient $\operatorname{HH}^*(\Lambda)/\mathcal{N}$ of the Hochschild cohomology ring by the ideal ${\mathcal N}$ generated by all homogeneous nilpotent elements is isomorphic to either $K$ or $K[x]$, and is thus finitely generated as an algebra. We also consider more generally the property of a finite dimensional algebra being selfinjective, and as a consequence show that if all simple $\Lambda$-modules are $\Omega$-periodic, then $\Lambda$ is selfinjective.