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On tau-tilting subcategories

Part of: Rings and algebras arising under various constructions Homological methods Abelian categories Homological algebra

Published online by Cambridge University Press:  08 March 2024

Javad Asadollahi Open the ORCID record for Javad Asadollahi [Opens in a new window] ,
Somayeh Sadeghi  and
Hipolito Treffinger
Javad Asadollahi*
Affiliation:
Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, 81746-73441 Isfahan, Iran e-mail: asadollahi@sci.ui.ac.ir
Somayeh Sadeghi
Affiliation:
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), 19395-5746 Tehran, Iran e-mail: somayeh.sadeghi@ipm.ir
Hipolito Treffinger
Affiliation:
Instituto de Investigaciones Matemáticas, Luis A. Santaló, UBA-CONICET, C1428 Buenos Aires, Argentina e-mail: htreffinger@dm.uba.ar
*
e-mail: asadollahi@ipm.ir
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Abstract

The main theme of this paper is to study $\tau $-tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $-cotorsion torsion triples and investigate a bijection between the collection of $\tau $-cotorsion torsion triples in $\mathscr {A}$ and the collection of support $\tau $-tilting subcategories of $\mathscr {A}$, generalizing the bijection by Bauer, Botnan, Oppermann, and Steen between the collection of cotorsion torsion triples and the collection of tilting subcategories of $\mathscr {A}$. General definitions and results are exemplified using persistent modules. If $\mathscr {A}=\mathrm{Mod}\mbox {-}R$, where R is a unitary associative ring, we characterize all support $\tau $-tilting (resp. all support $\tau ^-$-tilting) subcategories of $\mathrm{Mod}\mbox {-}R$ in terms of finendo quasitilting (resp. quasicotilting) modules. As a result, it will be shown that every silting module (resp. every cosilting module) induces a support $\tau $-tilting (resp. support $\tau ^{-}$-tilting) subcategory of $\mathrm{Mod}\mbox {-}R$. We also study the theory in $\mathrm {Rep}(Q, \mathscr {A})$, where Q is a finite and acyclic quiver. In particular, we give an algorithm to construct support $\tau $-tilting subcategories in $\mathrm {Rep}(Q, \mathscr {A})$ from certain support $\tau $-tilting subcategories of $\mathscr {A}$.

Keywords

Abelian category(τ-)tilting subcategorytorsion theorysilting modulequiver representation

MSC classification

Primary: 18E10: Exact categories, abelian categories 18E40: Torsion theories, radicals 16S90: Torsion theories; radicals on module categories
Secondary: 16E30: Homological functors on modules (Tor, Ext, etc.) 18G15: Ext and Tor, generalizations, Künneth formula
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 38
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The work of the first author is based on research funded by the Iran National Science Foundation (INSF) under Project No. 4001480. The research of the second author is supported by a grant from IPM. The third author is supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 893654.

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