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An integral degree-three Gersten counterexample

OpenAI · September 26, 2026

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Abstract

We construct a two-dimensional ramified regular local ring A in mixed characteristic $(0,5)$ for which the integral map $K_3(A)\to K_3(\mathop{\mathrm{Frac}}\nolimits A)$ has nonzero kernel. This gives a negative answer to the unrestricted integral Gersten conjecture.

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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, Local Gersten's conjecture for regular system of parameters (arXiv:1503.07966v8, August 2020). Its Theorem 5 is discussed in the companion degree-five paper's appendix as a vanishing claim that would kill the supported pushforward; this paper uses the same kind of pushforward for its $K_3$ class but does not cite the preprint.
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; this bears directly on Theorem 1.1's claimed failure of injectivity of $K_3(A)\to K_3(\mathrm{Frac}\,A)$.
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 mod-5 detection in $G_3(A[1/\pi];\mathbb Z/5)$, 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, including a $K_3$ class over a three-dimensional ramified ring, addressing the question left open here of whether the image of $c$ in $K_3(A)\otimes\mathbb Q$ is nonzero; its reference list does not cite the OpenAI papers. [AI-generated per source]

Elsewhere

Emergent Mind: family 209

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026,
  author = {{OpenAI}},
  title = {{An integral degree-three Gersten counterexample}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf}{OAI:An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026}},
  year = {2026}
}