In an abelian category $\mathscr{ A}$, we can generate torsion pairs from tilting objects of projective dimension $\leq 1$. However, when we look at tilting objects of projective dimension $2$, there is no longer a natural choice of an associated torsion pair. Instead of trying to generate a torsion pair, Jensen, Madsen and Su generated a triple of extension closed classes that can filter any objects of $\mathscr{ A}$. We generalize this result to proper abelian subcategories.
Categories: math.RT