Constructing large modules over artin algebras

Let be an artin algebra and denote by mod(() the category of nitely generated-modules. Apart from those we are also interested in-modules which are not nitely generated. These are called large. Recall that the algebra is of innnite representation type provided that mod(() has innnitely many isomorphism classes of indecomposable objects. A theorem of Auslander asserts that there exists a large indecomposable-module whenever is of innnite representation type 2]. In this note we use his method to construct speciic large indecomposable modules which arise naturally from certain innnite families of morphisms in mod(() having isomorphic kernels. We apply this result as follows. Suppose there is given a chain of monomorphisms between nitely generated indecomposable-modules. Denote by X 1 = ?! limX i the corresponding direct limit. It is natural to ask for conditions that ensure the indecomposability of X 1. In fact, chains of monomorphisms of the above form occur for quasi-serial components of the Auslander-Reiten quiver of. Ringel calls any module X = X s belonging to such a component quasi-serial of quasi-length s, since it determines, up to isomorphism, uniquely a chain of irreducible monomorphisms i : X i ! X i+1 , i 2 N, such that there exists no irreducible monomorphism ending in X 1 8]. We show that the corresponding direct limit X 1 = ?! limX i is a large indecomposable module. Moreover, we obtain an exact sequence where (Tr D) s (X) denotes the s-th power of the transpose of the dual of X which is again a quasi-serial module. This exact sequence has the following property. Every morphism : X ! Y in mod(() belongs to the innnite radical of mod(() if and only if factors through X , and moreover, induces a commutative diagram of the following form