∑math20

Algebra · Result 209

Integral counterexamples to Gersten’s conjecture

2 papers

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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Papers 2

First page of An integral counterexample to Gersten's conjecture

An integral counterexample to Gersten's conjecture

OpenAI · September 25, 2026

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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@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}
}
First page of An integral degree-three Gersten counterexample

An integral degree-three Gersten counterexample

OpenAI · September 26, 2026

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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@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}
}

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