On the stable module category of a self-injective algebra

Abstract

Let A be a finite-dimensional self-injective algebra. We study the dimensions of spaces of stable homomorphisms between indecomposable Amodules which belong to Auslander-Reiten components of the form ZA or ZAo / (rk). The results are applied to representations of finite groups over fields of prime characteristic, especially blocks of wild representation type. We are interested in homological properties of modules for self-injective algebras. An important invariant of a finite-dimensional algebra is its stable Auslander-Reiten quiver; and during the past years, it has played a crucial role for classification problems of self-injective algebras of finite or tame representation type. We call a stable component quasi-serial, if it is of the form ZAoo or ZAoo/(rk). If A is any tame algebra, then most of its Auslander-Reiten components are homogeneous tubes, by [CB], hence are quasi-serial. Moreover, it seems that for self-injective algebras of wild type most components of the stable Auslander-Reiten quiver are quasi-serial. For a block of a group algebra which is of wild representation type, all components are of this form (see [E2]). Therefore, it is important to understand homological properties of quasi-serial components. If A is a finite-dimensional self-injective algebra and X, Y are A-modules, then we denote the stable homomorphisms from X to Y by Hom(X, Y). Following the terminology in [RI], if M is a module in a quasi-serial stable component C, then the quasi-length of M is the number of the row to which M belongs. The module M is quasi-simple if it has quasi-length one, that is, if it lies at the end of the component. The first two chapters contain basic facts on dimensions of spaces of stable homomorphisms; this is intended to provide tools which may be of more general use. In Chapter 3 we prove the following general results. Suppose f is an equivalence of the stable module category of A, consider stable homomorphisms Hom(M, fM) for modules M in a quasi-serial component C. We prove that the dimension of Hom(M, fM) is weakly increasing as a function of the quasi-length of M, except possibly when TQ-1 f T on C (see 3.3). Moreover, we study the cases when f = IS and f = QS in more detail. We show that if C is a tube, then the dimensions of Hom(M, rSM) for M e C are unbounded (3.5). In general, if C is a component of the form ZA,, which is not fixed by Q and the dimensions of Hom(M, TsM) are bounded, then Hom(X, r-mX) 0 Received by the editors June 4, 1996 and, in revised form, October 2, 1997. 2000 Mathematics Subject Classification. Primary 18G25; Secondary 16G70, 20C20.