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

Edward L. Green, Nicole Snashall, and 0yvind Solberg

Abstract.
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 $\HH^*(\Lambda)/\N$ of the Hochschild cohomology ring
by the ideal $\N$ 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
self-injective, and as a consequence show that if all simple $\Lambda$-modules
are $\Omega$-periodic, then $\lambda$ is selfinjective.