$\tau$-perpendicular wide subcategories

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 paper, 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.

v3: final version. v2: Removed Corollary 1.2a and added discussion of a counterexample as Remark 6.17, corrected errors in the proof of Theorem 1.3, and made other small changes to organization and exposition. 22 pages, 2 figures

References in corpus (1)

• Rigid modules and ICE-closed subcategories in quiver representations

Cited by in corpus (1)

• The category of a partitioned fan