Hochschild cohomology of Beilinson algebra of exterior algebra

Let An be the Beilinson algebra of exterior algebra of an n-dimensional vector space, which is derived equivalent to the endomorphism algebra Endox （T） of a tilting complex T = II^ni=0^Ox （i） of coherent COx-modules over a projective scheme X = P^nk. In this paper we first construct a minimal projective bimodule resolution of An, and then apply it to calculate k-dimensions of the Hochsehild cohomology groups of An 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 An. As a consequence, we provide a class of algebras of finite global dimension whose Hochschild cohomology rings have non-trivial multiplicative structures.

Let An be the Beilinson algebra of exterior algebra of an n-dimensional vector space,which is derived equivalent to the endomorphism algebra End O X(T) of a tilting complex T = Πni=0 OX(i) of coherent O X-modules over a projective scheme X = P n k.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.