Hochschild cohomology of finite—dimensional algebras

Abstract

The aim of this article is to report on some recent results on computing Hochschild cohomology groups of finite-dimensional algebras. For this let k be an algebraically closed field and A a finite-dimensional k-algebra (associative, with unit). By modA we denote the category of finitely generated left A-modules. Let AXA be a finitely generated A-bimodule. The Hochschild cohomology groups Hi(A,X) (i 0) were introduced by Hochschild [Ho] (for a definition see section 1). The lowdimensional groups (i ::; 2) have a very concrete interpretation of classical algebraic structures such as derivations and extensions. It was observed by Gerstenhaber [GeJ that there are also connections to algebraic geometry. In fact, H2(A, A) controls the deformation theory of A. And it was shown that the algebras A which satisfy H2(A, A) = 0 are rigid. For a similar approach we also mention an article of Gabriel [Ga L]. Despite this very little was done in actual computations for particular classes of finitedimensional algebras. In section 1 we briefly review the fundamental definitions of Hochschild cohomology and include an alternative description which one often uses for direct computations. In section 2 we present some computations. This includes a report on results due to Cibils [elJ, [C2J, [C3J and Gerstenhaber and Schack [GSJ. For some of these results we have included proofs and some examples. In section 3 we deal with derivations. In the remaining two sections we outline how recently emerged methods in the representation theory of finite-dimensional algebras yield information on the Hochschild cohomology.