Tame and wild theorem for the category of filtered by standard modules

Abstract

We introduce the notion of interlaced weak ditalgebras and apply reduction procedures to their module categories to prove a tame-wild dichotomy for the category ${\cal F}(\Delta)$ of $\Delta$-filtered modules for an arbitrary finite homological system $({\cal P},\leq,\{\Delta_i\}_{i\in {\cal P}})$. This includes the case of standardly stratified algebras. 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 proofs rely on the equivalence of ${\cal F}(\Delta)$ with the module category of some special type of ditalgebra.