Generic Modules Over Artin Algebras

Generic modules have been introduced by Crawley-Boevey in order to provide a better understanding of nite dimensional algebras of tame representation type. In fact he has shown that the generic modules correspond to the one-parameter families of indecomposable nite dimensional modules over a tame algebra 5]. The Second Brauer-Thrall Conjecture provides another reason to study generic modules because the existence of a generic module over an artin R-algebra (R= rad R an innnite eld) implies that has strongly unbounded representation type, i.e. there are innnitely many pairwise non-isomorphic-modules of length n for innnitely many n 2 N 6]. The aim of this paper is to develop further the analysis of existence and properties of generic modules. Our approach depends to a large extent on the embedding of a module category into a bigger functor category. These general concepts are explained in the rst two sections. We continue in Section 3 with a new characterization of the pure-injective modules which occur as the source of a minimal left almost split morphism. This is of interest in our context because generic modules are pure-injective. Next we consider indecomposable endoonite modules. Recall that a module is endoonite if it is of nite length when regarded in the natural way as a module over its endomorphism ring. Changing slightly the original deenition, we say that a module is generic if it is indecomposable endoonite but not nitely presented. Section 4 is devoted to several characterizations of generic modules in order to justify the choice of the non-nitely presented modules as the generic objects. We prove them for dualizing rings, i.e. a class of rings which includes noetherian algebras and artinian PI-rings. Existence results for generic modules over dualizing rings follow in Section 5. Several results in this paper depend on the fact that a functor f: Mod(?) ! Mod(() which commutes with direct limits and products, preserves certain niteness conditions. For example, if a ?-module M is endoonite then f(M) is endoonite. If in addition End ? (M) is a PI-ring, then End (N) is a PI-ring for every indecomposable direct summand N of f(M). This material is collected in Section 6 and 7. In Section 8 we introduce an eeective method to construct generic modules over artin algebras from so-called generalized tubes. The special case of a tube in the Auslander-Reiten quiver is discussed in the following section. We obtain 1