∞-tilting theory

By L. Positselski, J. Št’ovíček. Pacific Journal of Mathematics, 2017

346 citations recorded by the source corpus.

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.

View Paper: https://www.semanticscholar.org/paper/68a6b6704deb69f37c160a1af04e5cd15a4f1b07

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Citations in the corpus

Silting Objects (2018, background): “The zero cohomologies of such complexes form an interesting class of modules, first studied systematically by Adachi, Iyama and Reiten in [3], which parametrise torsion pairs in module categories and are also related to ring theoretic localisation [14, 15].”
Minimal inclusions of torsion classes (2017, background): “In [1], the authors describe the lattice of functorially-finite torsion classes by way of τ -tilting pairs.”
Wide Subcategories and Lattices of Torsion Classes (2019, background): “The first named author of this paper introduced brick labeling in [2] for functorially finite torsion classes in the module category mod A in the context of the Koenig–Yang correspondences [11, 22] and τ -tilting theory [1].”
A reduction theorem for τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}-rigid modules (2016, background): “Introduction The theory of support τ -tilting modules, as introduced by Adachi, Iyama and Reiten in [4], is related to, and to some extent generalizes, several classical concepts in the representation theory of finite dimensional algebras.”