Does Tachikawa's second conjecture imply the Auslander-Reiten conjecture?
A family of homological conjectures for artin algebras - the Nakayama conjecture, the generalized Nakayama conjecture, the Auslander-Reiten conjecture, the Auslander-Gorenstein conjecture, the Gorenstein-projective conjecture and Tachikawa's first and second conjectures - has been known since the 1970s to sit in a web of one-way implications, with the question of which are equivalent left open. Does Tachikawa's second conjecture imply the Auslander-Reiten conjecture?
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Representation theory of artin algebras
- Posed by
- Maurice Auslander and Idun Reiten, in their work on the generalized Nakayama conjecture; Tachikawa's conjectures date from the same period
- Year posed
- 1975
- Years open
- 51y
- Solved
- 2026-09-14
- Model
- GPT-6 Astra; Claude Opus 5
- Vendor
- OpenAI; Anthropic
- Collaborators
- Haruhisa Enomoto
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Yes. With the known implications it follows that the Auslander-Reiten conjecture, the generalized Nakayama conjecture, the Auslander-Gorenstein conjecture, the Nakayama conjecture, the Gorenstein-projective conjecture and Tachikawa's second conjecture are all equivalent, and that Tachikawa's second implies his first. None of them is thereby proved.
What the AI did
From the paper's use-of-AI section: the proof was 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 revised the exposition; the author set the structure and takes responsibility.
Verification
Checked here on 22 September 2026 against arXiv:2609.19172: the abstract states the implication for artin algebras over a commutative artinian ring, by way of the two-fold trivial extension, and draws the consequence that six of the conjectures in the family become equivalent and that Tachikawa's second implies his first. The introduction confirms the Auslander-Reiten attribution. The mathematics was not checked here; eight days old, no referee. Same author and same AI process as the periodicity-conjecture entry of five days earlier.