$\infty$-tilting theory

Leonid Positselski and Jan Šťovíček

Vol. 301 (2019), No. 1, 297–334

DOI: 10.2140/pjm.2019.301.297

Abstract

We define the notion of an infinitely generated tilting object of infinite homological dimension in an abelian category. A one-to-one correspondence between $\infty$-tilting objects in complete, cocomplete abelian categories with an injective cogenerator and $\infty$-cotilting objects in complete, cocomplete abelian categories with a projective generator is constructed. We also introduce $\infty$-tilting pairs, consisting of an $\infty$-tilting object and its $\infty$-tilting class, and obtain a bijective correspondence between $\infty$-tilting and $\infty$-cotilting pairs. Finally, we discuss the related derived equivalences and t-structures.

Keywords

tilting theory, Wakamatsu tilting module, pseudo-derived equivalence

Mathematical Subject Classification 2010

Primary: 18E10, 18E30

Secondary: 16E30

Milestones

Received: 12 August 2018

Accepted: 29 September 2018

Published: 16 September 2019