Estimate of the number of one-parameter families of modules over a tame algebra Original Research Article

The problem of classifying modules over a finite dimensional algebra A reduces to a linear matrix problem whose indecomposable matrices can be parametrized by certain normal forms, called canonical matrices in [Linear Algebra Appl. 317 (2000) 53]. If the algebra A is tame, then the set of isomorphism classes of indecomposable A-modules of dimension at most d is given by a finite number f(d,A) of discrete modules and one-parameter families of modules. Accordingly, there are for every fixed size only a finite number of discrete and one-parameter families of canonical matrices. We prove that in the tame case the number of one-parameter families of canonical matrices of size m×n and a given partition into blocks is bounded by 4mn. Based on this estimate, we prove thatimagewherer is the number of nonisomorphic indecomposable projective A-modules and δ1,…,δr are their dimensions. This estimate improves significantly the double exponential estimate from [C. R. Acad. Sci. Paris 322 (Sèrie I) (1996) 211].