Abstract
AbstractLet
$\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.
Publisher
Cambridge University Press (CUP)