AI 中文总结
该研究针对有限维代数的两个倾斜完备化问题给出否定答案，构造反例反驳相关猜想，并证明两个自正交猜想间的等价关系。
AI 中文摘要
我们对有限维代数的两个倾斜完备化问题给出否定回答，构造了两个有限维、基础、连通的拟遗传$\boldsymbol{\textit{C}}$代数：第一个代数存在忠实、基础、满秩的预倾斜模，但无倾斜完备化；第二个代数存在几乎倾斜模，但无倾斜完备化。该满秩例子还为两个猜想提供了反例：Enomoto的自正交Wakamatsu-倾斜猜想，以及Chen、Li、Zhang和Zhao的自正交忠实猜想。我们进一步证明，自正交Wakamatsu-倾斜猜想对所有有限维代数成立当且仅当自正交忠实猜想对所有有限维代数成立。该构造最终源于Krah在有理曲面上长度为极大的非满例外集合，以及Kalck对应的满秩预silting例子。
英文摘要
We give negative answers to two tilting-completion questions for finite-dimensional algebras. We construct two finite-dimensional basic connected quasi-hereditary $\mathbb C$-algebras. The first admits a faithful basic full-rank pretilting module with no tilting completion; the second admits an almost-tilting module with no tilting completion. The full-rank example also yields counterexamples to two conjectures: Enomoto's Self-orthogonal Wakamatsu-tilting Conjecture and the Self-orthogonal Faithful Conjecture of Chen, Li, Zhang, and Zhao. We further show that the Self-orthogonal Wakamatsu-tilting Conjecture holds for all finite-dimensional algebras if and only if the Self-orthogonal Faithful Conjecture holds for all finite-dimensional algebras. The construction ultimately stems from Krah's non-full exceptional collection of maximal length on a rational surface and Kalck's associated full-rank presilting example.
Comments21 pages, comments are welcome!
