Triangulated categories with cluster-tilting subcategories

TL;DR

This paper introduces and studies a new class of subcategories called ghost cluster tilting subcategories in triangulated categories, establishing links with $ au$-tilting theory and generalizing existing results.

Contribution

It defines ghost cluster tilting subcategories, explores their properties, and connects them with $ au$-tilting theory, extending prior work on cluster tilting subcategories.

Findings

Established a bijection between weak $ [1]$-cluster tilting and support $ au$-tilting subcategories.

Characterized subcategories of $ ext{mod} $ corresponding to cluster tilting subcategories.

Proved equivalence between ghost cluster tilting objects and relative cluster tilting objects.

Abstract

Let $\C$ be a triangulated category with a cluster tilting subcategory $\T$. We introduce the notion of $\T[1]$-cluster tilting subcategories (also called ghost cluster tilting subcategories) of $\C$, which are a generalization of cluster tilting subcategories. We first develop a basic theory on ghost cluster tilting subcategories. Secondly, we study links between ghost cluster tilting theory and $\tau$-tilting theory: Inspired by the work of Iyama, J{\o}rgensen and Yang \cite{ijy}, we introduce the notion of $\tau$-tilting subcategories and tilting subcategories of $\mod\T$. We show that there exists a bijection between weak $\T[1]$-cluster tilting subcategories of $\C$ and support $\tau$-tilting subcategories of $\mod\T$. Moreover, we figure out the subcategories of $\mod\T$ which correspond to cluster tilting subcategories of $\C$. This generalizes and improves several results by Adachi-Iyama-Reiten \cite{AIR}, Beligiannis \cite{Be2}, and Yang-Zhu \cite{YZ}. Finally, we prove that the definition of ghost cluster tilting objects is equivalent to the definition of relative cluster tilting objects introduced by the first and the third author in \cite{YZ}.