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$\tau $-PERPENDICULAR WIDE SUBCATEGORIES

Part of: Representation theory of rings and algebras

Published online by Cambridge University Press:  22 August 2023

ASLAK BAKKE BUAN Open the ORCID record for ASLAK BAKKE BUAN [Opens in a new window]  and
ERIC J. HANSON Open the ORCID record for ERIC J. HANSON [Opens in a new window]
ASLAK BAKKE BUAN
Affiliation:
Department of Mathematical Sciences Norwegian University of Science and Technology (NTNU) 7491 Trondheim, Norway [email protected]
ERIC J. HANSON*
Affiliation:
Department of Mathematical Sciences Norwegian University of Science and Technology (NTNU) 7491 Trondheim, Norway
*
[email protected]
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Abstract

Let $\Lambda $ be a finite-dimensional algebra. A wide subcategory of $\mathsf {mod}\Lambda $ is called left finite if the smallest torsion class containing it is functorially finite. In this article, we prove that the wide subcategories of $\mathsf {mod}\Lambda $ arising from $\tau $-tilting reduction are precisely the Serre subcategories of left-finite wide subcategories. As a consequence, we show that the class of such subcategories is closed under further $\tau $-tilting reduction. This leads to a natural way to extend the definition of the “$\tau $-cluster morphism category” of $\Lambda $ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan–Marsh in the $\tau $-tilting finite case and by Igusa–Todorov in the hereditary case.

Keywords

τ-tilting theoryperpendicular categoryτ-tilting reductionτ-cluster morphism categories

MSC classification

Primary: 16G10: Representations of Artinian rings 16G20: Representations of quivers and partially ordered sets
Type
Article
Information
Nagoya Mathematical Journal , Volume 252 , December 2023 , pp. 959 - 984
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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Footnotes

This work was supported by the Norwegian Research Council (Grant No. FRINAT 301375).

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