arXivDaily arXiv每日学术速递 周一至周五更新
arXiv 1808.02709math.RT
收藏

子范畴中的$d$ -Auslander-Reiten序列

$d$-Auslander-Reiten sequences in subcategories

Francesca Fedele

备注：
27 pages. Final version, which has been accepted for publication in the Proceedings of the Edinburgh Mathematical Society
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

设$Φ$是域$k$上的有限维代数. Kleiner描述了Auslander-Reiten序列在一个前覆盖扩展闭子类$\mathcal{X}\subseteq$ mod $Φ$中。如果$X\in\mathcal{X}$是不可分解的使得Ext $_Φ^1(X,\mathcal{X})\neq 0$和$ζX$是$\mathcal{X}$的唯一不可分解的直和项-cover $g:Y\rightarrow D$ Tr $X$使得Ext $_Φ^1(X,ζX)\neq 0$，则在$\mathcal{X}$中存在形式为\begin{align*} ε: 0\rightarrow ζX\rightarrow X'\rightarrow X\rightarrow 0. \end{align*}的Auslander-Reiten序列。此外，当End $_Φ(X)$模通过投射分解的态射是除环时，Kleiner证明了每一个\begin{align*} δ: 0\rightarrow Y\rightarrow Y'\xrightarrowη X\rightarrow 0 \end{align*}形式的非分裂短正合序列都使得$η$在$\mathcal{X}$中几乎是右分裂的，并且$δ$沿着$g$的推出给出了以$X$结尾的mod $Φ$中的Auslander-Reiten序列。在本文中，我们给出了高维的概括。令$d\geq 1$为整数。一个$d$ -簇倾斜子类$\mathcal{F}\subseteq$ mod $Φ$扮演一个更高mod $Φ$的角色.这样的$\mathcal{F}$是一个$d$ -阿贝尔范畴，其中核和上核被$d$对象的复合体取代，短正合序列被$d+2$对象的复合体取代。对于$\mathcal{F}$的一个加性“$d$ -扩展闭”子类$\mathcal{X}$，我们给出了上述结果的更高版本。

英文摘要：

Let $Φ$ 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 $Φ$. If $X\in\mathcal{X}$ is an indecomposable such that Ext$_Φ^1(X,\mathcal{X})\neq 0$ and $ζX$ is the unique indecomposable direct summand of the $\mathcal{X}$-cover $g:Y\rightarrow D$Tr$X$ such that Ext$_Φ^1(X,ζX)\neq 0$, 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$_Φ(X)$ 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 $η$ is right almost split in $\mathcal{X}$, and the pushout of $δ$ along $g$ gives an Auslander-Reiten sequence in mod $Φ$ 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 $Φ$ plays the role of a higher mod $Φ$. 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 $\mathcal{F}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑