Torsion pairs and filtrations in abelian categories with tilting objects

Abstract

Given a noetherian abelian category $\mathcal{Z}$ of homological dimension two with a tilting object $T$, the abelian category $\mathcal{Z}$ and the abelian category of modules over $\text{End}(T{)}^{\text{op}}$ 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 $\mathcal{Z}$ has a three-step filtration by extension-closed subcategories. Finally, we generalise Jensen-Madsen-Su's filtration to a noetherian abelian category of any finite homological dimension.

14 pages

References in corpus (4)

• A functorial construction of moduli of sheaves
• Quiver representations in toric geometry
• Some examples of tilt-stable objects on threefolds
• Filtrations in abelian categories with a tilting object of homological dimension two