arXiv 1904.045992019-04math.RT

温柔代数上完全复形的例外圈

Exceptional cycles for perfect complexes over gentle algebras

Peng Guo, Pu Zhang

备注：: 29 pages

摘要：

引入了一个具有Serre对偶的三角范畴$\mathcal T$中的例外圈。Broomhead，D. Pauksztello和D. Ploog，对$\mathcal T$的全球结构有显著影响。本文证明了：如果$\mathcal T$是同伦类的，则任何例外$1$ -圈都是不可分解的且在口;并且任何例外$n$ -圈中的对象都有$n\ge 3$在口.令$A$是具有$A\ne k$的不可分解的温和$k$ -代数。弦复合体在口部的Hom空间是明确确定的。主要结果分类“几乎所有的”例外循环在$K^b(A\mbox{-}{\rm proj})$，使用特征组件和它们的AG-不变量，除了那些例外的$1$ -循环，这是带复合物。即，$K^b(A\mbox{-}{\rm proj})$的特征分量$C$的口在$C$中形成唯一的例外循环，直到等价关系$\approx$ ;如果$A$的口不是$A_3$类型，则给出所有的例外$n$ -循环在$K^b(A\mbox{-}{\rm proj})$中与$n\ge 2$直到$\approx$ ;一个弦复形是一个例外的$1$ -圈当且仅当它位于一个具有{ \rm AG}-不变量$(1, m)$的特征分量的口。然而，口部的带状复合体可能不是例外的$1$周期。

英文摘要：

Exceptional cycles in a triangulated category $\mathcal T$ with Serre duality, introduced by N. Broomhead, D. Pauksztello, and D. Ploog, have a notable impact on the global structure of $\mathcal T$. In this paper we show that if $\mathcal T$ is homotopy-like, then any exceptional $1$-cycle is indecomposable and at the mouth; and any object in an exceptional $n$-cycle with $n\ge 3$ is at the mouth. Let $A$ be an indecomposable gentle $k$-algebra with $A\ne k$. The Hom spaces between string complexes at the mouth are explicitly determined. The main result classifies "almost all" the exceptional cycles in $K^b(A\mbox{-}{\rm proj})$, using characteristic components and their AG-invariants, except those exceptional $1$-cycles which are band complexes. Namely, the mouth of a characteristic component $C$ of $K^b(A\mbox{-}{\rm proj})$ forms a unique exceptional cycle in $C$, up to an equivalent relation $\approx$; if the quiver of $A$ is not of type $A_3$, this gives all the exceptional $n$-cycle in $K^b(A\mbox{-}{\rm proj})$ with $n\ge 2$, up to $\approx$; and a string complex is an exceptional $1$-cycle if and only if it is at the mouth of a characteristic component with {\rm AG}-invariant $(1, m)$. However, a band complex at the mouth is possibly not an exceptional $1$-cycle.

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。

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