Representation Type and Stable Equivalence of Morita Type for Finite Dimensional Algebras

In this note we show that two nite dimensional algebras have the same representation type if they are stably equivalent of Morita type. Stable equivalences of Morita type were introduced for blocks of group algebras by Brou e 2], see also 7]. The concept is motivated by a result of Rickard. In 9], he proved that any derived equivalence between nite dimensional self-injective algebras and ? induces a stable equivalence of a particulary nice form. More precisely, he constructs bimodules B ? and ? C such that the corresponding tensor functors induce mutually inverse equivalences between the stable categories mod and mod ? of nite dimensional modules for and ?. Stable equivalences of this form are said to be of Morita type. They occur frequently, for instance in the theory of blocks of group algebras. Other examples are obtained from two nite dimensional algebras and ? which are tilted from each other. The corresponding trivial extensions T and T? are derived equivalent self-injective algebras 8], and therefore stably equivalent of Morita type. In fact, Assem and de la Pe~ na proved in 1] that T and T? have the same representation type; however their methods are completely diierent from the approach presented in this paper. The representation type of a nite dimensional algebra is traditionally deened using the concept of a continuous one-parameter family in the category mod of nite dimensional-modules. For instance, is of tame representation type if for every n 2 N only nitely many such families are needed to parametrize all indecom-posable-modules of dimension n. An alternative approach uses so-called generic modules. This was suggested by Crawley-Boevey and he established for tame algebras a correspondence between continuous one-parameter families and generic modules 3]. In 6], we used generic modules to show that a stable equivalence mod ! mod ? induces a bijection between continuous one-parameter families in mod and mod ?. However, without any extra assumption on the stable equivalence it remains an open question how the dimensions of the modules in mod and mod ? are related. In this paper we settle the problem for stable equivalences which are induced by an appropriate functor mod ! mod ?. Stable equivalences of Morita type are of this form, and therefore we can prove that two nite dime-sional algebras have the same representation type if they are stably equivalent of Morita type. I would like to thank Thorsten …