Tame and wild theorem for the category of filtered by standard modules for a quasi-hereditary algebra

Abstract

We introduce the notion of interlaced weak ditalgebras and apply reduction procedures to their module categories to prove the tame-wild dichotomy for the category ${\cal F}(\Delta)$ of filtered by standard modules for a quasi-hereditary algebra. Moreover, in the tame case, we show that given a fixed dimension $d$, for every $d$-dimensional indecomposable module $M \in {\cal F}(\Delta)$, with the only possible exception of those lying in a finite number of isomorphism classes, the module $M$ coincides with its Auslander-Reiten translate in ${\cal F}(\Delta)$. Our results are based on a theorem by Koenig, K\"ulshammer, and Ovsienko relating ${\cal F}(\Delta)$ with the module category of some special type of ditalgebra.Comment: 51 page