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

Abstract

This paper describes the Hochschild cohomology ring of a selfinjective algebra Λ \Lambda of finite representation type over an algebraically closed field K K , showing that the quotient HH ∗ ⁡ ( Λ ) / N \operatorname {HH}^*(\Lambda )/\mathcal {N} of the Hochschild cohomology ring by the ideal N {\mathcal N} generated by all homogeneous nilpotent elements is isomorphic to either K K or K [ x ] 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 Λ \Lambda -modules are Ω \Omega -periodic, then Λ \Lambda is selfinjective.