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Universal test rings for rational Gersten injectivity

A mix of human-written and AI-generated textHuman understanding: all partsmath.KT — K-Theory and Homologymath.AG — Algebraic Geometry

Contributed by Niels FELD

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

For each prime pp, rational Gersten injectivity in a fixed degree for mixed-characteristic henselian discrete valuation rings of residue characteristic pp is equivalent to injectivity for W(Fp(t1,t2,…)‾)W(\overline{\mathbb{F}_p(t_1,t_2,\ldots)}). For a finite bound dd on residue transcendence degree, the test ring is W(Fp(t1,…,td)‾)W(\overline{\mathbb{F}_p(t_1,\ldots,t_d)}). For these henselian DVRs, rational localization boundary images depend only on the residue field and are reflected by arbitrary residue-field extensions. A normalization by the absolute ramification index makes the resulting comparisons of kernels canonical and functorial. The proof uses regular approximation, finite transfers, and perfection. An appendix gives an alternative comparison based on the existential embedding theorem of Anscombe--Dittmann--Jahnke, including an example with nonnoetherian valuation rings. We also describe the additional boundary condition that induction between consecutive test rings would require.

Provenance statement

The main mathematical ideas were found during discussions with chatgpt (GPT-5.5 and then Sol 5.6). The paper was mostly written by Sol 5.6 and GPT-6 Astra, under strict supervision. Separate AI-assisted (GPT-6 Astra and Opus 5.5) referee passes were also used to search for proof gaps and incorrect hypotheses. The author assumes full responsibility for the mathematical statements, proofs, and references.

Tools used

Anthropic
Claude OpusVersion 5.5
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ChatGPTVersion 5.6
OpenAI
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OpenAI
ChatGPTVersion 6

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Version history

  1. v1Submitted by Niels FELDInitial depositSupersededOct 28, 2026
  2. v2Submitted by Niels FELDSubmitted as a new versionCurrentOct 29, 2026