The periodicity conjecture for finite-dimensional algebras
A module is periodic when it is isomorphic to one of its own higher syzygies, and an algebra is periodic when it is periodic as a bimodule over itself. The periodicity conjecture, studied by Erdmann and Skowroński in their work on periodic and weighted surface algebras, asks whether a finite-dimensional algebra must be periodic whenever all of its simple modules are. Is it true?
- Result
- Disproved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Representation theory of algebras
- Posed by
- Karin Erdmann and Andrzej Skowroński, in their programme on periodic algebras and blocks of group algebras
- Year posed
- 2015
- Years open
- 11y
- Solved
- 2026-09-09
- Model
- GPT-6 Astra; Claude Opus 5
- Vendor
- OpenAI; Anthropic
- Collaborators
- Haruhisa Enomoto
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
False: an explicit 36-dimensional algebra whose simple modules all have period four, but which has a 2-dimensional nonperiodic module and so is not itself periodic.
What the AI did
From the paper's use-of-AI section: the counterexample and its proof were found by GPT-6-Astra in research directed by the author, and the first draft of the manuscript was written by GPT-6-Astra; Claude Opus 5 reviewed the exposition; the author set the structure and takes responsibility.
Verification
Checked here on 22 September 2026 against arXiv:2609.09732: the abstract states the conjecture and gives a 36-dimensional counterexample whose simple modules have period four and which carries a 2-dimensional nonperiodic module, so the algebra itself is not periodic. The reference list confirms the Erdmann-Skowroński attribution (Colloq. Math. 138 (2015) on the periodicity conjecture for blocks of group algebras; J. Algebra 505 (2018) on weighted surface algebras). The mathematics was not checked here; thirteen days old, no referee.