{"schema":"postcutoff/event@1","as_of":"2026-10-14T07:26:00+02:00","url":"https://postcutoff.com/e/2026-10-11-skryabin-conjecture-noetherian-hopf-algebra-antipode/","md":"https://postcutoff.com/e/2026-10-11-skryabin-conjecture-noetherian-hopf-algebra-antipode/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-11-skryabin-conjecture-noetherian-hopf-algebra-antipode","date":"2026-10-11","date_precision":"day","short_title":"Skryabin's 2006 conjecture proved","deck":"Every noetherian Hopf algebra has a bijective antipode (proof strategy suggested by GPT-6)","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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.15473: A noetherian Hopf algebra has bijective antipode","url":"https://arxiv.org/abs/2610.15473","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-14","updated":"2026-10-14","orgs":["Université libre de Bruxelles","OpenAI"],"title":"Skryabin's 2006 conjecture proved: every noetherian Hopf algebra has a bijective antipode (proof strategy suggested by GPT-6)","summary":"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.'"],"key_numbers":[],"tags":["math","algebra","hopf-algebras","conjecture","gpt-6"],"science":{"field":"mathematics","subfield":"Hopf algebras / noncommutative ring theory","problem":"Skryabin's conjecture: every noetherian Hopf algebra has a bijective antipode","result":"Proved, in the stronger form: finite uniform dimension on one side implies a bijective antipode.","open_since":"2006","ai_system":["GPT-6"],"human_role":"AI-assisted: GPT-6 suggested the proof strategy in dialogue; the author developed, checked and wrote the proof","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint"},"status":"pending","shock":""},"body_md":"## What happened\n\nOn 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.\n\nThe \"Use of AI\" note credits GPT-6 with the proof strategy, reached \"through an iterative dialogue with the author\".\n\nNo expert reactions were found on 14 Oct (web search; X not searched directly).\n\n## Why it matters\n\nBijectivity 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.","disputed":[{"label":"Verification","text":"Unrefereed preprint","sources":[]}],"related":[{"id":"2026-10-13-ai-assisted-results-october-2026","url":"https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/","date":"2026-10-13","date_precision":"day","short_title":"October 2026: the arXiv stream of conjectures settled with disclosed AI help continues","deck":null,"takeaway":"In the first nine days of October 2026 the stream did not slow down: each arXiv listing brought several named conjectures with a disclosed AI role, and two new patterns appeared.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":16,"official":16,"filed":"2026-10-13","updated":"2026-10-14","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-14","type":"filed","text":"Created from the arXiv paper"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-14","run":null,"sources_read":"The arXiv paper","updated":"2026-10-14","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":164,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":103,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":194,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":133,"in_training_data":false}],"short_url":null}