FILTERING SUBCATEGORIES OF MODULES OF AN ARTINIAN ALGEBRA
Let A be an artinian algebra, and letC be a subcategory of modA that is closed under extensions. When C is closed under kernels of epimor- phisms (or closed under cokernels of monomorphisms), we describe the small- est class of modules that lters C. As a consequence, we obtain sucient conditions for the nitistic dimension of an algebra over a eld to be nite. We also apply our results to the torsion pairs. In particular, when a torsion pair is induced by a tilting module, we show that the smallest classes of mod- ules that lter the torsion and torsion-free classes are completely compatible with the quasi-equivalences of Brenner and Butler.