Relative rigid subcategories and $\tau$-tilting theory

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 $\mathrm{mod}(\mathcal{R}/\mathcal{P})$. 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.

arXiv comment21 pages. arXiv admin note: text overlap with arXiv:2003.12788

References in corpus (1)

• Relative rigid objects in extriangulated categories