Let Φ be a finite-dimensional algebra over a field k. Kleiner described the Auslander–Reiten sequences in a precovering extension closed subcategory ${\rm {\cal X}}\subseteq {\rm mod }\,\Phi $. If $X\in \mathcal {X}$ is an indecomposable such that ${\rm Ext}_\Phi ^1 (X,{\rm {\cal X}})\ne 0$ and $\zeta X$ is the unique indecomposable direct summand of the $\mathcal {X}$-cover $g:Y\to D\,{\rm Tr}\,X$ such that ${\rm Ext}_\Phi ^1 (X,\zeta X)\ne 0$, then there is an Auslander–Reiten sequence in $\mathcal {X}$ of the form $${\rm \epsilon }:0\to \zeta X\to {X}^{\prime}\to X\to 0.$$Moreover, when ${\rm En}{\rm d}_\Phi (X)$ modulo the morphisms factoring through a projective is a division ring, Kleiner proved that each non-split short exact sequence of the form $$\delta :0\to Y\to {Y}^{\prime}\buildrel \eta \over \longrightarrow X\to 0$$is such that η is right almost split in $\mathcal {X}$, and the pushout of δ along g gives an Auslander–Reiten sequence in ${\rm mod}\,\Phi $ ending at X.In this paper, we give higher-dimensional generalizations of this. Let $d\geq 1$ be an integer. A d-cluster tilting subcategory ${\rm {\cal F}}\subseteq {\rm mod}\,\Phi $ plays the role of a higher ${\rm 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 $\mathcal {F}$.

中文翻译：

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子类别中的 d-Auslander-Reiten 序列

设 Φ 是域上的有限维代数ķ. Kleiner 在预覆盖扩展封闭子类别中描述了 Auslander-Reiten 序列${\rm {\cal X}}\subseteq {\rm mod }\,\Phi $. 如果$X\in \mathcal {X}$是不可分解的，使得${\rm Ext}_\Phi ^1 (X,{\rm {\cal X}})\ne 0$和$\zeta X$是唯一的不可分解的直接和$\数学{X}$-覆盖$g:Y\to D\,{\rm Tr}\,X$这样${\rm 分机}_\Phi ^1 (X,\zeta X)\ne 0$，则有一个 Auslander-Reiten 序列$\数学{X}$形式的$${\rm \epsilon }:0\to \zeta X\to {X}^{\prime}\to X\to 0.$$此外，当${\rm En}{\rm d}_\Phi (X)$模射通过射影分解的态射是一个除环，Kleiner 证明了形式的每个非分裂短精确序列$$\delta :0\to Y\to {Y}^{\prime}\buildrel \eta \over \longrightarrow X\to 0$$是这样的η是对的，几乎分裂$\数学{X}$，并且推出δ沿着G给出 Auslander-Reiten 序列${\rm 模组}\,\Phi $结束于X.在本文中，我们对此进行了更高维的概括。让$d\geq 1$是一个整数。一种d-集群倾斜子类别${\rm {\cal F}}\subseteq {\rm mod}\,\Phi $扮演着更高的角色${\rm 模组}\,\Phi $. 这样一个$\数学{F}$是一个d-abelian 范畴，其中核和共核被d对象和由复合物组成的短精确序列d+ 2 个对象。对于添加剂'，我们给出了上述结果的更高版本d-扩展关闭'子类别$\数学{X}$的$\数学{F}$.