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Torsion in K1K_1 and pp-adic vanishing cycles

Primarily AI-generated textHuman understanding: all partsmath.AG — Algebraic Geometrymath.KT — K-Theory and Homology

Contributed by Akhil Mathew ↗

SubmitterAkhil Mathew

Version 1 / Sep 27, 2026 / CC BY 4.0

Abstract

For every odd prime pp, we construct a regular strictly henselian local ring AA whose generic fibre has \'etale cohomology classes in $H^2_{\et}(A[1/p],\mu_p^{\otimes2})$ which are not sums of cup products of degree-one classes. The ring is the strict henselization of a local ring on a finite-type arithmetic scheme. A unit on a principal divisor gives an element of order pp in SK1(A[1/p])SK_1(A[1/p]), detected by its localization boundary on the special fibre. Adams--Riemann--Roch and the connectivity of the motivic filtration show that this element has nonzero image in $H^3_{\mot}(A[1/p],\Z(2))$. The integral coefficient sequence then gives the cohomological obstruction. Only the eigenvalues 44 and 88 of ψ2\psi^2 are needed.

Provenance statement

The argument and writeup is entirely due to ChatGPT. I went through the argument and believe it to be correct. I also asked Claude to audit the argument and proofread the paper.

References

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Version history

  1. v1Initial depositCurrentSep 27, 2026