Δ-tame quasi-hereditary algebras

Abstract

Let (K, M, H) be an upper triangular biomodule problem. Brustle and Hille showed that the opposite algebra A of the endomorphism algebra of a projective generator P of the matrices category of (K, M, H) is quasi-hereditary, and there is an equivalence between the category of Δ-good modules of A and Mat(K, M). In this note, based on the tame theorem for bimodule problems, we show that if the algebra A associated with an upper triangular bimodule problem is of Δ-tame representation type, then the category F(Δ) has the homogeneous property, i.e. almost all modules in F(Δ) are isomorphic to their Auslander-Reiten translations. Moreover, if (K, M, H) is an upper triangular bipartite bimodule problem, then A is of Δ-tame representation type if and only if F(Δ) is homogeneous.