Addendum and Erratum: Mapping cones for morphisms involving a band complex in the bounded derived category of a gentle algebra
İlke Çanakçi Department of Mathematics, VU, Amsterdam, Amsterdam, 1081 HV, The Netherlands. , 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 note we correct two oversights in [Mapping cones in the bounded derived category of a gentle algebra, J. Algebra 530 (2019), 163–194] which only occur when a band complex is involved. As a consequence we see that the mapping cone of a morphism between two band complexes can decompose into arbitrarily many indecomposable direct summands.
中文速览
温和代数（gentle algebra）的有界导出范畴中，态射的映射锥（mapping cone）分解问题此前在一篇2019年的论文里已给出组合公式，但当涉及带复形（band complex）时存在两处疏漏：一是没有考虑"重叠段比同伦带本身更长"的图映射和拟图映射情形，二是错误地断言两个带复形之间态射的映射锥仍是一个不可分解的带复形。本文通过引入"展开图"（unfolded diagram）的精细分析和一个关键引理，系统地补全和纠正了这些情形下的映射锥分解公式。修正后的结论揭示出一个出人意料的现象：两个带复形之间的态射，其映射锥通常会分解成任意多个不可分解直和项，而非此前所称的单一带复形。这一结果不仅完善了温和代数上模的Ext空间的显式描述，也对Fukaya范畴中映射锥演算的组合解释具有重要意义。
原文 arXiv:2001.06435;中英对照 + 大白话阅读 https://aha.fim.ai/paper/2001.06435v1