$d$-Auslander-Reiten sequences in subcategories

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${}_{\Phi }^1(X,\mathcal{X})\ne 0$ and $\zeta X$ is the unique indecomposable direct summand of the $\mathcal{X}$-cover $g:Y\to D$Tr$X$ such that Ext${}_{\Phi }^1(X,\zeta X)\ne 0$, then there is an Auslander-Reiten sequence in $\mathcal{X}$ of the form \begin{align*} ε: 0\rightarrow ζX\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*} δ: 0\rightarrow Y\rightarrow Y'\xrightarrowη 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 $d\ge 1$ 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 $\mathcal{F}$.

27 pages. Final version, which has been accepted for publication in the Proceedings of the Edinburgh Mathematical Society