Let A be a tame hereditary algebra (finite-dimensional over an algebraically closed field), R™ (m > 1) the extension algebra of A. A generic jR-module M over an arbitrary ring R is by definition an indecomposable .R-module of infinite length, such that M considered as an End(M)module, is of finite length (its endolength). In this paper we investigate the generic modules of A (the repetitive algebra of A) and R™. It is proved that R™ has at least 2m generic modules.Introduction. The notion of generic module was introduced in [1] by Crawley-Boevey. The concept seems to be quite natural and important. The generic modules even have a dominating position in the category of modules. In [2], it was shown that whether a finite-dimensional algebra over an algebraically closed field is tame or wild is determined completely by the behaviour of the generic modules for that algebra.In [3], Aronszajn and Fixman gave the concept of a divisible module for the Kronecker algebra and showed that for the Kronecker algebra there exists a unique indecomposable torsion-free divisile module. In [4], Ringel generalized the work of Aronszajn and Fixman and proved the same result for a tame hereditary algebra. Ringel's work, in fact, showed that for a tame hereditary algebra, there exists a unique generic module. In [6], we solved the existence and uniqueness of generic module for the tilted algebra determined by a tame hereditary algebra.Following [1], A generic i?-module M over an arbitrary ring R is by definition an indecomposable .R-module of infinite length, such that M considered as an End(M)module, is of finite length (its endolength). Of course, the generic modules with endomorphism ring a division ring just, form the vertices of the (Cohn) spectrum of R. By [1], the endomorphism ring of a generic module always is a local ring.Our purpose here is to investigate the generic module of the extension algebra R™ (defined below) for a tame hereditary algebra A. In section 1, we investigate the z/-orbits of generic modules for a repetitive algebra, we shall prove that Mod A has at least two zA-orbits of generic ^-modules (Theorem 1.2). In section 2, we shall prove our main result on generic modules of R™: it^1 has at least 2m generic modules (Theorem 2.4 and Corollary 2.5).Throughout this paper, we denote by k an algebraically closed field. An algebra means basic, connected and finite-dimensional A:-algebra. For an algebra A we denote by Mod A the category of all right A-modules, by mod A the full subcategory of Mod A consisting of all finitely generated right ^4-modules and by mod A the corresponding stable category. We shall use freely properties of the Auslander-Reiten sequences, irreducible maps, Auslander-Reiten translation r = DTr and r -1 = TrD, and the Auslander-Reiten quiver TA of an algebra B, for which we refer to [5].
