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Here we show that, given a finite homological system $({\calP},\leq,\{\Delta_u\}_{u\in {\cal P}})$ for a finite-dimensional algebra$\Lambda$ over an algebraically closed field, the category ${\cal F}(\Delta)$of $\Delta$-filtered modules is tame if and only if, for any $d\in \mathbb{N}$,there are only finitely many isomorphism classes of generic $\Lambda$-modulesadapted to ${\cal F}(\Delta)$ with endolength $d$. We study the relationshipbetween these generic modules and one-parameter families of indecomposables in${\cal F}(\Delta)$. This study applies in particular to the category offiltered by standard modules for standardly stratified algebras.

Generic modules for the category of filtered by standard modules