Relative rigid subcategories and $\tau$-tilting theory

TL;DR

This paper explores the relationships between support τ-tilting subcategories, relative rigid subcategories, and cluster tilting in extriangulated categories, establishing correspondences and special cases that deepen understanding of their structure.

Contribution

It establishes a one-to-one correspondence between support τ-tilting subcategories of an abelian quotient and maximal relative rigid subcategories, advancing τ-tilting theory in extriangulated categories.

Findings

Correspondence between support τ-tilting and maximal relative rigid subcategories

Support tilting subcategories are a special case of support τ-tilting

Relation between tilting subcategories and cluster tilting when R is cluster tilting

Abstract

Let $\mathcal B$ be an extriangulated category with enough projectives $\mathcal P$ and enough injectives $\mathcal I$, and let $\mathcal R$ be a contravariantly finite rigid subcategory of $\mathcal B$ which contains $\mathcal P$. We have an abelian quotient category $\mathcal H/\mathcal R\subseteq \mathcal B/\mathcal R$ which is equivalent ${\rm mod}(\mathcal R/\mathcal P)$. In this article, we find a one-to-one correspondence between support $\tau$-tilting (resp. $\tau$-rigid) subcategories of $\mathcal H/\mathcal R$ and maximal relative rigid (resp. relative rigid) subcategories of $\mathcal H$, and show that support tilting subcategories in $\mathcal H/\mathcal R$ is a special kind of support $\tau$-tilting subcategories. We also study the relation between tilting subcategories of $\mathcal B/\mathcal R$ and cluster tilting subcategories of $\mathcal B$ when $\mathcal R$ is cluster tilting.