Infinite‐Dimensional Modules Over Wild Hereditary Algebras

One should be interested to know in which way the structure of modules of finite length determines the behaviour of arbitrary modules. Recall that a finite-dimensional algebra is said to be of finite representation type if there are only finitely many indecomposable modules of finite length. Then any module is the direct sum of modules of finite length (Ringel-Tachikawa, 1974), and such a decomposition is unique up to isomorphism. M. Auslander has shown (in “Large modules over artin algebras”, 1976) that if A is not of finite representation type, then there exist indecomposable modules which are not of finite length. Auslander gave an existence proof and C. M. Ringel gave a general structure theory for modules of arbitrary length in his “Rome Lectures” (1977, published 1979 [R3]). He showed that there always will be certain important infinite-dimensional representations, and the investigation of these modules also gives some new insight into the behaviour of the modules of finite length. Note also that in general one cannot dualize results for arbitrary modules, since the dual functor D = Homk(−, k) is only an equivalence between the categories mod−A and A − mod. Most definitions are motivated by the structure theory of C. M. Ringel for the tame hereditary case, but there is not enough time to present the details for infinite-dimensional A-modules, when A is tame. [R3] is worth reading.