We define a support variety for a finitely generated module over an artin algebra Λ over a commutative artinian ring k, with Λ flat as a module over k, in terms of the maximal ideal spectrum of the algebra HH*(Λ) of Λ. This is modelled on what is done in modular representation theory, and the varieties defined in this way are shown to have many of the same properties as those for group rings. In fact the notions of a variety in our sense and those for principal and non‐principal blocks are related by a finite surjective map of varieties. For a finite‐dimensional self‐injective algebra over a field, the variety is shown to be an invariant of the stable component of the Auslander–Reiten quiver. Moreover, we give information on nilpotent elements in HH*(Λ), give a thorough discussion of the ring HH*(Λ) on a class of Nakayama algebras, give a brief discussion of a possible notion of complexity, and make a comparison with support varieties for complete intersections. 2000 Mathematics Subject Classification 16E40, 16G10, 16P10, 16P20 (primary), 13D03, 16G70, 20J06 (secondary).

Abstract. Support varieties for any finite dimensional algebra over a field were introduced in [20] using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb's theorem is true.

We investigate the Drinfeld doubles D( n,d ) of a certain family of Hopf algebras. We determine their simple modules and their indecomposable projective modules, and we obtain a presentation by quiver and relations of these Drinfeld doubles, from which we deduce properties of their representations, including the Auslander-Reiten quivers of the D( n,d ). We then determine decompositions of the tensor products of most of the representations described, and in particular give a complete description of the tensor product of two simple modules. This study also leads to explicit examples of Hopf bimodules over the original Hopf algebras.

Let Λ = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution F of the graded simple modules over Λ is given in [9]. This resolution is shown to have a "comultiplicative" structure in [7], and this is used to find a minimal projective resolution P of Λ over the enveloping algebra Λ e . Using these results we show that the multiplication in the Hochschild cohomology ring of Λ relative to the resolution P is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of Λ with respect to a canonical basis over k associated to the resolution F. The natural map from the Hochschild cohomology to the Koszul dual of Λ is shown to be surjective onto the graded centre of the Koszul dual.