Skryabin’s 2006 conjecture proved
Every noetherian Hopf algebra has a bijective antipode (proof strategy suggested by GPT-6)
Awaiting review
The takeaway
A single-author 8-page paper settles a well-known question about Hopf algebras; the author says the proof strategy came from an iterative dialogue with GPT-6.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 14 October 2026
Your AI and this story
- GPT-6 Astra164 days after its cutoff
- Claude Opus 5.5103 days after its cutoff
- Gemini 4 Argon194 days after its cutoff
- Grok 4.7133 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 103 days before it.
Key facts
- Paper: ‘A noetherian Hopf algebra has bijective antipode’, Andrea Sciandra (Université Libre de Bruxelles), arXiv 2610.15473 [math.RA], 11 Oct 2026
- Theorem: if either regular module of a Hopf algebra H (over an arbitrary field) has finite uniform dimension, the antipode S is bijective; corollaries: left and right uniform dimensions agree, and H is right noetherian iff left noetherian
- Conjecture: Skryabin (2006), every noetherian Hopf algebra has a bijective antipode; the proof starts from Skryabin’s injectivity theorem for weakly finite Hopf algebras
- AI use: ‘The proof strategy was suggested by GPT-6 (OpenAI) through an iterative dialogue with the author; intermediate results helped identify the key difficulties and arrive at a simpler argument. All AI-assisted material used in this paper has been checked by the author, who takes full responsibility for the content of the paper.’
What happened
On 11 Oct 2026 Andrea Sciandra posted arXiv 2610.154731. Skryabin had shown that the antipode of a weakly finite Hopf algebra is injective and conjectured in 2006 that noetherian Hopf algebras have bijective antipodes. Sciandra proves more: one-sided finite uniform dimension is enough. The proof builds “essential left ideals of common denominators” for finite-dimensional subcoalgebras and uses an inverse-antipode identity for surjectivity.
The “Use of AI” note credits GPT-6 with the proof strategy, reached “through an iterative dialogue with the author”.
No expert reactions were found on 14 Oct (web search; X not searched directly).
Why it matters
Bijectivity of the antipode for noetherian Hopf algebras was a standard open question in the area. It is another named conjecture where a frontier model supplied the strategy and a single human author did the rest.
What is disputed or not yet verified
| Verification | Unrefereed preprint |
|---|
Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed from the arXiv paper