Hochschild cohomology of Beilinson algebra of exterior algebra

Abstract

Let Λn be the Beilinson algebra of exterior algebra of an n-dimensional vector space, which is derived equivalent to the endomorphism algebra \(End_{\mathcal{O}_X } (T)\) of a tilting complex T = Πi=0nOX(i) of coherent OX-modules over a projective scheme X = Pkn. In this paper we first construct a minimal projective bimodule resolution of Λn, and then apply it to calculate k-dimensions of the Hochschild cohomology groups of Λn in terms of parallel paths. Finally, we give an explicit description of the cup product and obtain a Gabriel presentation of Hochschild cohomology ring of Λn. As a consequence, we provide a class of algebras of finite global dimension whose Hochschild cohomology rings have non-trivial multiplicative structures.