Hochschild Cohomology Of A Class Of Quantized Koszul Algebras

Snashall and Solberg introduced support variety theory to finitely generated mod-ules of a finite dimensional k-algebra by Hochschild cohomology in2004, and also conjectured that the Hochschild cohomology ring modulo nilpotence (i.e. HH*(A)/N) of a finite dimensional k-algebra A is always finitely generated as algebra. The con-jecture is proved to be ture for many classes of algebras. Until2008, Xu F. gave the first counterexample of the conjecture by studying the Hochschild cohomology of category algebras and computing HH*(A-1)/N of a Kozsul algebra A-1in the case chark=2. In this paper, we mainly compute the dimensions of Hochschild cohomology spaces of a quantized class of algebras Aq(qΣ k\{0}), describe the cup product of Hochschild cohomology of Aq, and therefore determine the multiplicative structures of Hochschild cohomology rings of them. Finally, we determine the structure of HH*(Aq)/N, and obtain more counterexamples of Snashall-Solberg conjecture.Firstly, we construct the minimal projective bimodule resolution of Aq, and by ex-plicit analysis over parameter q and the characteristic of k, we describe the dimensions of Hochschild cohomology spaces.Secondly, by defining the comultiplication structure, we show the cup product of Hochschild cohomology of Aq is essentially the juxtaposition of parallel path up to coefficient, and thus determine the multiplicative structures of Hochschild cohomology rings of Aq. Then we determine the structure of HH*(Aq)/)N, and thus prove that HH*(Aq)/N is not finitely generated as algebras if q is root of unity.