---
id: "2026-10-11-skryabin-conjecture-noetherian-hopf-algebra-antipode"
url: "https://postcutoff.com/e/2026-10-11-skryabin-conjecture-noetherian-hopf-algebra-antipode/"
as_of: "2026-10-14T07:26:00+02:00"
date: "2026-10-11"
date_precision: day
category: science
importance: 3
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 1
editor: human editor
human_review: null
version: null
---

As of: 2026-10-14 07:26 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-11-skryabin-conjecture-noetherian-hopf-algebra-antipode/

# Skryabin's 2006 conjecture proved

Full title: Skryabin's 2006 conjecture proved: every noetherian Hopf algebra has a bijective antipode (proof strategy suggested by GPT-6)

arXiv 2610.15473 (11 Oct 2026) by Andrea Sciandra proves that a Hopf algebra over any field has a bijective antipode as soon as its left or right regular module has finite uniform dimension; in particular every noetherian Hopf algebra has a bijective antipode, which was conjectured by S. Skryabin in 2006. The "Use of AI" note says "The proof strategy was suggested by GPT-6 (OpenAI) through an iterative dialogue with the author". Unrefereed preprint.

## 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 2610.15473](https://arxiv.org/abs/2610.15473). 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 164 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 103 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 194 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 133 days after its cutoff

## Sources

1. [arXiv 2610.15473: A noetherian Hopf algebra has bijective antipode](https://arxiv.org/abs/2610.15473) (arxiv.org, paper)

## Changes

- 2026-10-14 (filed): Created from the arXiv paper

## Related

- 2026-10-13: [October 2026: the arXiv stream of conjectures settled with disclosed AI help continues](https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/index.md)