Vol. 301, No. 1, 2019

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$\infty$-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 -tilting objects in complete, cocomplete abelian categories with an injective cogenerator and -cotilting objects in complete, cocomplete abelian categories with a projective generator is constructed. We also introduce -tilting pairs, consisting of an -tilting object and its -tilting class, and obtain a bijective correspondence between -tilting and -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
Authors
Leonid Positselski
Institute of Mathematics
Czech Academy of Sciences
Prague
Czech Republic
Laboratory of Algebraic Geometry
National Research University Higher School of Economics
Moscow
Russia
Sector of Algebra and Number Theory
Institute for Information Transmission Problems
Moscow
Russia
Department of Mathematics
Faculty of Natural Sciences
University of Haifa
Mount Carmel
Israel
Jan Šťovíček
Department of Algebra
Faculty of Mathematics and Physics
Charles University in Prague
Praha
Czech Republic