The First Cohomology Group of an Algebra with Coefficients in a Bimodule

Abstract

Ž . After their introduction in the 1940s cf. 18 , the Hochschild homology and cohomology groups of an algebra with coefficients in a bimodule have played a fundamental role. Given an algebra and a -bimodule M and viewing both and M as left modules over the enveloping algebra e op Ž . , the Hochschild cohomology groups H* , M are the Ž . Ž . groups Ext* , M while the homology groups H , M are the correŽ . sponding Tor groups cf. 5, Chap. IX . The most studied case has been that in which M and, more specifically, when is a split basic finite dimensional algebra over a field K. In that case, the algebra is partially determined by combinatorial data, namely, its quiver with relations, and it Ž . Ž Ž .. is a natural goal to obtain information about H* , M resp. H , M in terms of those combinatorial data. That was the scheme of the initial attempts made by Cibils 6 and Happel 17 to give explicit formulae for Ž . the dimensions of the H* , . In terms of computations, the first natural situation to look at is that in which the algebra is monomial, since