Is a finite locally free group scheme killed by its order?

Determine the function assigning to each positive integer n the least positive integer m such that every finite locally free group scheme of rank n, over every commutative ring, is killed by m. The value is 0 if there is no such m, since sInf ∅ = 0 in ℕ.

Current state (tracked automatically)

Formal Conjectures marks the Lean statement optimal_exponent as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `optimal_exponent`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 4 solved	Formal Conjectures

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Cite

@misc{cairn-paper-finite-locally-free-group-scheme-exponent,
  title        = {Is a finite locally free group scheme killed by its order?},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-finite-locally-free-group-scheme-exponent}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

Determine the function assigning to each positive integer $n$ the least positive integer $m$ such that every finite locally free group scheme of rank $n$ , over every commutative ring, is killed by $m$ . The value is $0$ if there is no such $m$ , since sInf ∅ = 0 in ℕ. As in killed_by_rank, the rings and algebras are in Type.

Grothendieck asked whether this least exponent is $n$ itself. It is not, already for $n = 4$ . If some positive exponent kills every group scheme of rank $n$ , then these exponents are the multiples of the least one, so minimality for $\leq$ agrees with minimality for divisibility.

A finite locally free group scheme of rank n is the group scheme analogue of a finite group of order n. Lagrange's theorem gives g ^ n = 1 in a finite group of order n, and Grothendieck asked whether the n-th power map of such a group scheme is trivial. The answer is affirmative over a reduced base and for commutative group schemes, and negative in general. Which exponents kill every group scheme of a given rank is open.

A group scheme is represented here by its coordinate Hopf algebra A over the base ring R. A is not assumed to be cocommutative, so the group scheme is not assumed to be commutative. The base is a ring rather than a scheme: a finite locally free group scheme is affine over its base, and being killed by m is local on the base, so nothing is lost.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.FiniteLocallyFreeGroupSchemeExponent.

theorem optimal_exponent :
    ∀ n, sInf ((↑) '' {m | IsUniversalKillingExponent.{0, 0} n m} : Set ℕ) = optimal_exponent.answer
      n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[M. Demazure and A. Grothendieck, Schémas en groupes (SGA 3), re-edition by P. Gille and P. Polo](https://webusers.imj-prg.fr/~patrick.polo/SGA3/). Exp. VIII, Rem. 7.3.1 poses the question; an editors' note records that the re-edition drops the commutativity assumption there. Exp. VIIA, Prop. 8.5 settles the case of a field and Cor. 8.5.2 the case of a reduced base.
[R. Schoof, *Is a finite locally free group scheme killed by its order?*](https://reneschoof.github.io/schoof_oortAAG.pdf), a survey of the question and of the cases that are known, written before the counterexample
J. Tate and F. Oort, Group schemes of prime order, p. 4 for Deligne's theorem and p. 5, which records the general question as open in 1970
[Mathlib, *A finite free group scheme of rank four that is not killed by four*](https://leanprover-community.github.io/mathlib4_docs/Counterexamples/GrothendieckPower.html), the counterexample

Source and licence

Imported from Formal Conjectures (research papers), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.