$d$-Auslander-Reiten sequences in subcategories

Francesca Fedele

Published 2018-08-08Version 1

Let $\Phi$ be a finite dimensional algebra over an algebraically closed field. Kleiner described the Auslander-Reiten sequences in a precovering extension closed subcategory $\mathcal{X}\subseteq\text{mod }\Phi$. If $X\in\mathcal{X}$ is an indecomposable such that $\text{Ext}^1(X,\mathcal{X})\neq 0$ and $\sigma X$ is the unique indecomposable direct summand of the $\mathcal{X}$-cover $g:Y\rightarrow D\text{Tr } X$ such that $\text{Ext}^1(X,\sigma X)\neq 0$, then there is an Auslander-Reiten sequence in $\mathcal{X}$ of the form \begin{align*} \epsilon: 0\rightarrow \sigma X\rightarrow X'\rightarrow X\rightarrow 0. \end{align*} Moreover, when $\text{End } (X)$ 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 $\text{mod }\Phi$ 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 $\mathcal{F}\subseteq\text{mod }\Phi$ plays the role of a higher $\text{mod }\Phi$. 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 $\mathcal{F}$.