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Grothendieck's Finite Flat Group Scheme Order Question

Algebra · posed by Alexander Grothendieck, 1966 · disproved

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Statement

Grothendieck asked whether every finite locally free group scheme of order nn is killed by nn (its nn-th convolution power map equals the unit). The counterexample is an order-4 group scheme not killed by 4 (killed only by 8); since Deligne settled the commutative case, it is necessarily non-commutative over a non-reduced base.

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No person has examined this. Everything below was judged by machines. say whether it holds →

  1. construction · #1

    GPT-5.6 Sol, Claude Fable 5, with Akhil Mathew and Kevin Buzzard

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    OpenAI's Sol found an explicit counterexample, a rank-4 Hopf algebra over Z[a,b]/(a3,b3,a2b+2)\mathbb{Z}[a,b]/(a^3, b^3, a^2 b + 2) whose order-4 group scheme is not killed by 4, and Claude Fable 5 autoformalized the full argument in Lean within hours. Akhil Mathew directed the work and submitted it to Mathlib; Kevin Buzzard independently compiled and checked the 1076-line proof.

  2. Machine-checked by Lean on #1 · not a person

    lean: correctLean

    scope Lean formalization of the result

    Machine-checked in Lean and submitted to Mathlib (PR #41748, opened 2026-07-14, disclosed as built with OpenAI's Codex and Anthropic's Claude under the author's direction). Kevin Buzzard independently compiled the 1076-line proof and confirmed it uses only standard mathlib definitions. Under active expert review (Wieser, Brasca) and not yet merged; no journal publication yet, but the counterexample is explicit and kernel-checked.

    Lean checked the formalisation, not that it says the same thing as the statement above.

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open in Claude →open in ChatGPT →

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