$d$-Auslander-Reiten sequences in subcategories

TL;DR

This paper extends the theory of Auslander-Reiten sequences to higher-dimensional settings using $d$-cluster tilting subcategories, generalizing classical results to $d$-abelian categories with complex structures.

Contribution

It introduces higher-dimensional analogues of Auslander-Reiten sequences within $d$-cluster tilting subcategories, expanding the classical theory to $d$-abelian categories.

Findings

Established higher Auslander-Reiten sequences in $d$-abelian categories.

Generalized classical results to $d$-extension closed subcategories.

Provided structural descriptions of sequences in higher-dimensional contexts.

Abstract

Let $\Phi$ be a finite dimensional algebra over a field $k$. Kleiner described the Auslander-Reiten sequences in a precovering extension closed subcategory $\mathcal{X}\subseteq$ mod $\Phi$. If $X\in\mathcal{X}$ is an indecomposable such that Ext$_{\Phi}^1(X,\mathcal{X})\neq 0$ and $\zeta X$ is the unique indecomposable direct summand of the $\mathcal{X}$-cover $g:Y\rightarrow D$Tr$X$ such that Ext$_{\Phi}^1(X,\zeta X)\neq 0$, then there is an Auslander-Reiten sequence in $\mathcal{X}$ of the form \begin{align*} \epsilon: 0\rightarrow \zeta X\rightarrow X'\rightarrow X\rightarrow 0. \end{align*} Moreover, when End$_\Phi (X)$ modulo the morphisms factoring through a projective is a division ring, Kleiner proved that each non-split short exact sequence of the form \begin{align*} \delta: 0\rightarrow Y\rightarrow Y'\xrightarrow{\eta} X\rightarrow 0 \end{align*} is such that $\eta$ is right almost split in $\mathcal{X}$, and the pushout of $\delta$ along $g$ gives an Auslander-Reiten sequence in mod $\Phi$ ending at $X$. In this paper, we give higher dimensional generalisations of this. Let $d\geq 1$ be an integer. A $d$-cluster tilting subcategory $\mathcal{F}\subseteq$ mod $\Phi$ plays the role of a higher mod $\Phi$. Such an $\mathcal{F}$ is a $d$-abelian category, where kernels and cokernels are replaced by complexes of $d$ objects and short exact sequences by complexes of $d+2$ objects. We give higher versions of the above results for an additive "$d$-extension closed" subcategory $\mathcal{X}$ of $\mathcal{F}$.