Finitely generated modules over pullback rings and algebras

Classification of finitely generated indecomposable modules over a finite dimensional algebra, in terms of string and band modules, dates back at least to (Szkeres 1949). Combinatorial techniques of (Gelfand, Ponomarev 1968) have been put in a functorial setting (Ringel 1975), as summarized in (Butler, Shahzamanian 1980). These techniques are generalized in (Crawley-Boevey 1989) and used to characterize indecomposable representations of clans and, consequently, finitely generated indecomposable modules over clannish algebras.