Integral counterexamples to Gersten’s conjecture

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Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic \((0,5)\) have nonzero integral K-theory classes that vanish over their fraction fields.

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What the papers state

Theorem 1.1

In An integral degree-three Gersten counterexample, 26 September 2026

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The ring A is a two-dimensional Noetherian regular local domain of mixed characteristic (0,5), with \pi\in\mathfrak m_A^2. There is an integral class c\in K_3(A) such that

c|_{\Frac A}=0, \qquad c|_{A[1/\pi]}\bmod5\ne0 \quad\text{in }G_3(A[1/\pi];\Z/5).

In particular, K_3(A)\to K_3(\Frac A) is not injective.

Theorem 1.1

In An integral counterexample to Gersten’s conjecture, 25 September 2026

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Let V be the ring of integers in the unramified extension of \Q_5 of degree eight, and put

A=\bigl(V[x,y]/(5+x^4+y^4)\bigr)_{(5,x,y)}.

Then A is a Noetherian regular local domain of dimension two and

\ker\bigl(K_5(A)\longrightarrow K_5(\Frac A)\bigr)\ne0.

Consequently the unrestricted integral Gersten conjecture has a negative answer.

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Papers

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An integral counterexample to Gersten’s conjecture

25 September 2026, current edition.

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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.

An integral degree-three Gersten counterexample

26 September 2026, current edition.

Read it here PDF

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.

Lean statements

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Independent comparisons and proof-verification reports

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What people have said

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An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.
Author not stated. Source

VibeMathed, VibeMathed: math problems solved with AI, 11 October 2026. Source
Contested: conflicts with a published claim by Mochizuki (arXiv:1503.07966) on the local Gersten conjecture.

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