One-Parameter Families of Modules for Tame Algebras and Bocses

Abstract

Let Λ be a finite-dimensional algebra over an algebraically closed field k. We denote by mod Λ the category of finitely generated left Λ-modules. Consider the family ℱ(u) of the indecomposables M∈mod Λ such that \(\mathrm{dim}_{k}\,\mathrm{Hom}_{\Lambda}(M,\mathrm{Dtr}\,M)/\mathcal{S}(M,\mathrm{Dtr}\,M)=u\) , where \(\mathcal{S}(M,\mathrm{Dtr}\,M)\) is the subspace of morphisms which factorize through semisimple modules. If P,Q are projectives in mod Λ, ℱ(u)(P,Q) is the family of those modules M∈ℱ(u) such that a minimal projective presentation is of the formfM: P→Q. We prove that if Λ is of tame representation type then each ℱ(P,Q) has only a finite number of isomorphism classes or is parametrized by μ(u,P,Q) one-parameter families. We give an upper bound for this number in terms of u,P and Q. Then we give some sufficient conditions for tame of polynomial growth type. For the proof we consider similar results for bocses.