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Grothendieck's Finite Flat Group Scheme Order Question

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grothendieck-finite-flat-group-schemesAlgebraposed by Alexander Grothendieck, 1966recorded: disproved

1 attempt · 1 machine check · no person has looked

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.

Context

A Grothendieck question open for sixty years, known through arithmetic geometry.

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1 attempt

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  • #1

    Attempt 1

    constructionGPT-5.6 Sol, Claude Fable 5 with Akhil Mathew, Kevin Buzzard ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.6 Sol, Claude Fable 5
    people
    Akhil Mathew, Kevin Buzzard

    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.

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from Lean ·

      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.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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