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Frobenius orbit maps and rank-four group schemes

Primarily AI-generated textHuman understanding: some partsmath.AG — Algebraic Geometry

Contributed by Akhil Mathew ↗

SubmitterAkhil Mathew

Version 1 / Sep 27, 2026 / CC BY 4.0

Abstract

We construct finite free group schemes of rank four and exponent eight as stabilizers in a smooth two-dimensional affine group. We first compute the complete local deformation ring of the invariant space of constants and squares in characteristic two, using six Grassmannian coordinates. For every Artinian specialization, evaluation at the identity has a finite free rank-four stabilizer. Its fourth-power morphism is (x,y)↦(2bxy,2axy)(x,y)\mapsto(2bxy,2axy), and its eighth-power morphism is trivial. The fourth power is nonzero on the third-order neighbourhood of the universal deformation ring and on a specialization over Z[a,b]/(a2b−2,a3,b3)\Z[a,b]/(a^2b-2,a^3,b^3). We give complete formulas for the latter group scheme. Right translation preserves the chosen quadratic space exactly when it contains the square of the scalar character. This identifies (2a,2b)(2a,2b) as the common obstruction to fourth-power vanishing, normality of the stabilizer, and two-sidedness in an equivalent cyclic-module construction.

Provenance statement

The text was entirely generated by ChatGPT 6 Pro. I also used Claude to help with auditing the paper and checking the formulas. In preparing this, I also allowed Codex and Claude to use computer tools such as Macaulay2 and Singular.

References

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  9. A rank-four counterexample to Grothendieck's power question , source manuscript in the repository GrothendieckRankP2 , snapshot of 4 August 2026, commit 79ccdd4 . https://github.com/j2d9w5xtjn-png/GrothendieckRankP2/blob/79ccdd4a479ef1e1a8a252337547bf3c810efa0b/manuscripts/main/A_RANK_FOUR_COUNTEREXAMPLE_TO_GROTHENDIECKS_POWER_QUESTION_2026-07-12.tex Versioned source manuscript , consulted 26 September 2026. The date in the filename is not the date of this snapshot.link
  10. The Mathlib Community, A finite free group scheme of rank four that is not killed by four , Lean source module Counterexamples/GrothendieckPower.lean , in Mathlib , 2026, repository snapshot 040b7f43a9ed . https://github.com/leanprover-community/mathlib4/blob/040b7f43a9edc2407eb03eb197fdece2b9b5b5d4/Counterexamples/GrothendieckPower.lean Versioned Lean source ; https://leanprover-community.github.io/mathlib4_docs/Counterexamples/GrothendieckPower.html generated module documentation , consulted 26 September 2026. The cited declarations are in the namespace Counterexample.GrothendieckPower .link
  11. DOI 10.1007/978-1-4612-1974-3DOI

Version history

  1. v1Initial depositCurrentSep 27, 2026