Mapping cones in the bounded derived category of a gentle algebra
İlke Çanakçi School of Mathematics, Statistics and Physics, Newcastle University, Newcastle Upon Tyne, NE1 7RU, United Kingdom. , David Pauksztello Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, United Kingdom and Sibylle Schroll Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, United Kingdom.
Abstract
In this article we describe the triangulated structure of the bounded derived category of a gentle algebra by describing the triangles induced by the morphisms between indecomposable objects in a basis of their Hom-space.
中文速览
温和代数（gentle algebra）的有界导出范畴（bounded derived category）里，不可分解对象之间的扩张中间项长什么样，这一问题此前并无一般性的完整描述。本文利用同伦弦（homotopy string）与同伦带（homotopy band）的组合语言，系统计算了标准基态射（图映射、单映射、双映射）的映射锥（mapping cone），从而明确给出了任意两个弦复形或带复形之间扩张的中间项分解。核心结果以一套直观的图形演算呈现：只需把两条同伦弦/带按重叠部分拼在一起，再从特定位置"切出"红框和绿框，便能读出映射锥的所有不可分解直和项。这一结果意义深远，不仅直接回答了模范畴中Ext¹基的一个长期开放问题，还为丛理论中的交换三角、三角范畴中的挠对分类以及温和代数导出范畴的几何模型提供了关键工具。
原文 arXiv:1609.09688;中英对照 + 大白话阅读 https://aha.fim.ai/paper/1609.09688v4