  Let $A$ be an artinian algebra, and let $\mathcal{C}$ be a subcategory of
mod$A$ that is closed under extensions. When $\mathcal{C}$ is closed under
kernels of epimorphisms (or closed under cokernels of monomorphisms), we
describe the smallest class of modules that filters $\mathcal{C}$. As a
consequence, we obtain sufficient conditions for the finitistic dimension of an
algebra over a field to be finite. 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 modules that filter the torsion and
torsion-free classes are completely compatible with the quasi-equivalences of
Brenner and Butler.
