τ-PERPENDICULAR WIDE SUBCATEGORIES

Abstract

Let Λ be a finite-dimensional algebra. A wide subcategory of modΛ is called left finite if the smallest torsion class containing it is functorially finite. In this article, we prove that the wide subcategories of modΛ arising from τ -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 τ -tilting reduction. This leads to a natural way to extend the definition of the “ τ -cluster morphism category” of Λ to arbitrary finite-dimensional algebras. This category was recently constructed by Buan–Marsh in the τ -tilting finite case and by Igusa–Todorov in the hereditary case.