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 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 selfinjective, and as a consequence show that if all simple Lambda-modules are Omega-periodic, then Lambda is selfinjective.