Grothendieck's Finite Flat Group Scheme Order Question
Grothendieck asked whether every finite locally free group scheme of order n is killed by n (its n-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.
- Result
- Disproved
- Field
- Algebraic Geometry
- Posed by
- Alexander Grothendieck
- Year posed
- 1966
- Years open
- 60y
- Solved
- 2026-07-11
- Model
- GPT-5.6 Sol, Claude Fable 5
- Vendor
- OpenAI / Anthropic
- Collaborators
- Akhil Mathew, Kevin Buzzard
- Verification
- Lean-verified
- Notability
- No dedicated article
What the AI did
OpenAI's Sol found an explicit counterexample, a rank-4 Hopf algebra over 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.
Verification
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.