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An integral counterexample to Gersten's conjecture

OpenAI · September 25, 2026

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Abstract

We construct a two-dimensional ramified regular local ring A of mixed characteristic $(0,5)$ for which $K_5(A)\to K_5(\mathop{\mathrm{Frac}}\nolimits A)$ has a nonzero kernel. This disproves Gersten's conjecture in its unrestricted integral form.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Possibly relevant literature (found by the agent, not checked)

S. Mochizuki, Gersten's conjecture (arXiv:0704.2275, April 2007). A short preprint whose abstract claims to prove Gersten's conjecture for every commutative regular local ring, which bears directly on Theorem 1.1's claim that the unrestricted integral conjecture fails; the paper's bibliography cites only the later Mochizuki preprint 1503.07966.
S. Mochizuki, A survey of Gersten's conjecture (arXiv:1608.08114, August 2016). Survey notes on the conjecture and on the author's motivic version, relevant to the history paragraph and to the appendix on Mochizuki's Lemma 13(III).
N. Feld, Finite-coefficient K-theory of henselian valued fields and Gersten injectivity (arXiv:2608.18789, August 2026). Proves finite-coefficient Gersten injectivity for noetherian henselian regular local rings with invertible coefficients; a point of comparison for the paper's mod-5 generic noninjectivity, where 5 is the residue characteristic.
Morten Lüders and Matthew Morrow, Milnor K-theory of p-adic rings (arXiv:2101.01092, 4 January 2021). Their mod-$p$-power Gersten theorem for Milnor K-theory, Nisnevich-locally on smooth schemes over complete discrete valuation rings, is a comparison point for the family's distinction between smooth finite-coefficient results and ramified integral Quillen K-theory kernels. [family-level]
HayamaKazuma (GitHub), Rational Gersten kernels in ramified regular local rings (GitHub repository, accessed 2026-10-21). Claims infinite-order classes in rational Gersten kernels for ramified regular local rings, the rational variant this paper leaves open, and gives an obstruction to Mochizuki's Lemma 13(III); its reference list does not cite the OpenAI papers. [AI-generated per source]

Elsewhere

VibeMathed: Problem record
Emergent Mind: family 209

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026,
  author = {{OpenAI}},
  title = {{An integral counterexample to Gersten's conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf}{OAI:An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026}},
  year = {2026}
}