Relative cluster tilting theory and $\ddot{I}„$-tilting theory

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 $\ddot{I}„$-tilting theory in functor categories and abelian categories.

35 pages.In this version, we have added many new results