Torsion pairs and filtrations in abelian categories with tilting objects

Given a noetherian abelian k-category Z of finite homological dimension, with a tilting object T of projective dimension 2, the abelian category Z and the abelian category of modules over End(T)(op) are related by a sequence of two tilts; we give an explicit description of the torsion pairs involved. We then use our techniques to obtain a simplified proof of a theorem of Jensen-Madsen-Su, that Z has a three-step filtration by extension-closed subcategories. Finally, we generalize Jensen-Madsen-Su's filtration to the case where T has any finite projective dimension.