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PreprintCC BY 4.0arXiv

A proper dg algebra which does not cogenerate

  • Mingyuan Hu
  • Vivek Shende
  • Dinglong Wang
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{hu2026proper,
  title = {{A proper dg algebra which does not cogenerate}},
  author = {Mingyuan Hu and Vivek Shende and Dinglong Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.34193},
  eprint = {2609.34193},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.34193},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A proper dg algebra which does not cogenerate
AU  - Mingyuan Hu
AU  - Vivek Shende
AU  - Dinglong Wang
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.34193
UR  - https://arxiv.org/abs/2609.34193
AB  - 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.
ER  -