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

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Added 1 Nov 2026 · not yet checked against the literature · importance 0.50

What the prompt says

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Statement

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 $\le$ agrees with minimality for divisibility.

Lean statement

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

Sources(1)

formal-conjectures: FormalConjectures/Paper/FiniteLocallyFreeGroupSchemeExponent.lean (opens in a new tab)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/Paper/FiniteLocallyFreeGroupSchemeExponent.lean.

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