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.