AI 中文总结
本文记录了一个不余生成其模范畴的共连通 dg 代数例子，并附带相关现象，其存在性关联于 Weinstein 流形对 Arnol'd 弦猜想的违反。
AI 中文摘要
Keller 的同调猜想的强形式断言：任何有限维代数都余生成其无界导出模范畴。这里我们记录一个（共连通的）dg 代数的例子，其上有有限维上同调，但不余生成其模范畴，并附带一些相关现象：一个光滑 dg 范畴，其对偶双模不是非退化的，以及一个非平凡的完全忠实左 Calabi-Yau 态射。所有例子和大部分证明均由 ChatGPT 生成。在附录中，我们解释了我们寻找此类例子的原因：它们的存在将遵循 Weinstein 辛流形的存在性，而这些流形不满足 Arnol'd 的弦猜想。
英文摘要
Keller's strong form of the homological conjectures asserts: any finite dimensional algebra cogenerates its unbounded derived category of modules. Here we record an example of a (coconnective) dg algebra with finite dimensional cohomology, which does not cogenerate its module category, along with some related phenomena: a smooth dg category whose dualizing bimodule fails to be nondegenerate, and a nontrivial fully faithful left Calabi-Yau morphism. All examples and most proofs were produced by ChatGPT. In an appendix we explain the reason we were looking for such examples: their existence would follow from the existence of Weinstein symplectic manifolds which failed to satisfy Arnol'd's chord conjecture.
