Relative cluster tilting theory and $\tau$-tilting theory

TL;DR

This paper explores the mutation properties of two-term weak cluster tilting subcategories in triangulated categories and applies these findings to $ au$-tilting theory in functor and abelian categories.

Contribution

It establishes the mutation behavior of almost complete two-term weak $ ext{R}[1]$-cluster tilting subcategories and extends the results to $ au$-tilting theory.

Findings

Almost complete two-term weak $ ext{R}[1]$-cluster tilting subcategories have exactly two completions.

Results connect relative cluster tilting theory with $ au$-tilting theory.

Applications to functor and abelian categories.

Abstract

Let $\mathcal C$ be a Krull-Schmidt triangulated category with shift functor $[1]$ and $\mathcal R$ be a rigid subcategory of $\mathcal C$. We are concerned with the mutation of two-term weak $\mathcal R[1]$-cluster tilting subcategories. We show that any almost complete two-term weak $\mathcal R[1]$-cluster tilting subcategory has exactly two completions. Then we apply the results on relative cluster tilting subcategories to the domain of $\tau$-tilting theory in functor categories and abelian categories.