Almost split morphisms in subcategories of triangulated categories

TL;DR

This paper explores the structure of subcategories in triangulated categories, focusing on almost split morphisms, Ext-projectives, and their role in mutations and cluster categories, extending classical Auslander-Reiten theory.

Contribution

It introduces a new perspective on almost split morphisms in subcategories, linking them to mutations and extension-closed subcategories in triangulated and cluster categories.

Findings

Characterizes when a subcategory admits a triangle similar to an Auslander-Reiten triangle.

Shows that replacing an Ext-projective object with a related object produces a new extension-closed subcategory.

Provides a full description of such triangles in cluster categories of Dynkin type A_n.

Abstract

For a suitable triangulated category $\mathcal{T}$ with a Serre functor $S$ and a full precovering subcategory $\mathcal{C}$ closed under summands and extensions, an indecomposable object $C$ in $\mathcal{C}$ is called Ext-projective if Ext$^1(C,\mathcal{C})=0$. Then there is no Auslander-Reiten triangle in $\mathcal{C}$ with end term $C$. In this paper, we show that if, for such an object $C$, there is a minimal right almost split morphism $\beta:B\rightarrow C$ in $\mathcal{C}$, then $C$ appears in something very similar to an Auslander-Reiten triangle in $\mathcal{C}$: an essentially unique triangle in $\mathcal{T}$ of the form \begin{align*} \Delta= X\xrightarrow{\xi} B\xrightarrow{\beta} C\rightarrow \Sigma X, \end{align*} where $X$ is an indecomposable not in $\mathcal{C}$ and $\xi$ is a $\mathcal{C}$-envelope of $X$. Moreover, under some extra assumptions, we show that removing $C$ from $\mathcal{C}$ and replacing it with $X$ produces a new subcategory of $\mathcal{T}$ closed under extensions. We prove that this process coincides with the classic mutation of $\mathcal{C}$ with respect to the rigid subcategory of $\mathcal{C}$ generated by all the indecomposable Ext-projectives in $\mathcal{C}$ apart from $C$. When $\mathcal{T}$ is the cluster category of Dynkin type $A_n$ and $\mathcal{C}$ has the above properties, we give a full description of the triangles in $\mathcal{T}$ of the form $\Delta$ and show under which circumstances replacing $C$ by $X$ gives a new extension closed subcategory.