受保护的角与韩氏猜想的三分法
Protected corners and a trichotomy for Han's conjecture
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中文总结 AI 辅助
研究韩氏猜想，在τ-霍赫希尔德框架下，证明受保护角定理界定生存顶点处的Ext，解决相关扩展猜想，还建立间隙A失败在三个强连通顶点上的三分法，包括排除全无限情形及分类两无限情形为相互哑铃形。
中文摘要 AI 辅助
韩氏猜想预测具有最终消失的霍赫希尔德同调的有限维代数具有有限的整体维数。在西比尔斯、兰齐洛塔、马科斯和索洛塔尔的τ-霍赫希尔德框架中，它分为持久性（间隙A）和生存性（间隙B），间隙A的失败已经是一个反例。我们证明了一个受保护角定理，界定了生存顶点处的Ext，解决了单项式和特殊双列情形之外的刘-莫林扩展猜想。然后，我们在三个强连通顶点上建立了间隙A失败的三分法：全无限情形不可能，两无限情形完全分类为相互哑铃形。
英文摘要
Han's conjecture predicts that a finite-dimensional algebra with eventually vanishing Hochschild homology has finite global dimension. In the tau-Hochschild framework of Cibils, Lanzilotta, Marcos and Solotar, it splits into persistence (Gap A) and survival (Gap B), and a Gap-A failure is already a counterexample. We prove a protected corner theorem bounding Ext at a surviving vertex, settling the Liu-Morin extension conjecture beyond the monomial and special biserial cases. We then establish a trichotomy for Gap-A failures on three strongly connected vertices: the all-infinite case is impossible, and the two-infinite case is completely classified as a mutual dumbbell.
