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$\tau $-PERPENDICULAR WIDE SUBCATEGORIES
- Part of
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- Journal:
- Nagoya Mathematical Journal / Volume 252 / December 2023
- Published online by Cambridge University Press:
- 22 August 2023, pp. 959-984
- Print publication:
- December 2023
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- Article
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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.
