Result 209, Algebra

Integral counterexamples to Gersten’s conjecture

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

Disproof or counterexample

The bigger picture

Why it matters

Allowing division by every nonzero element can erase algebraic information that a major conjecture predicted would survive. Two manuscripts report explicit examples of this loss, identifying a limit to recovering information about a ring from its fraction field.

What changes?

The manuscripts construct two explicit two-dimensional ramified regular local rings, algebraic models of well-behaved neighborhoods of points. Both have mixed characteristic (0,5): the ring has characteristic zero, while its residue field has characteristic five. Their integral K-theory groups, which encode algebraic structure, contain nonzero classes that become zero in the fraction field, where all nonzero elements are invertible. One construction concerns degree 5, the other degree 3. These claimed failures of injectivity disprove Gersten's conjecture in its unrestricted integral form.

What does that help mathematicians do?

For these rings and degrees, knowing a K-theory class after passage to the fraction field is not enough to determine whether it was zero originally. The examples therefore rule out an unrestricted principle for detecting integral K-theory classes this way. Researchers seeking valid injectivity statements must account for these cases through appropriate restrictions. The claims do not establish failure in every degree or for every regular local ring.

Are there practical applications?

The immediate value is foundational: these examples identify information that can be lost when replacing a local ring by its fraction field, a simpler setting where division is available. They give researchers concrete counterexamples against which to check proposed integral K-theory statements. The supplied abstracts describe no practical application beyond this mathematical role.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

An integral counterexample to Gersten's conjecture

September 25, 2026 25 pages

We construct a two-dimensional ramified regular local ring A of mixed characteristic (0,5)(0,5) for which K5(A)→K5(FracA)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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Cite (BibTeX) 
@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}
}

An integral degree-three Gersten counterexample

September 26, 2026 10 pages

We construct a two-dimensional ramified regular local ring A in mixed characteristic (0,5)(0,5) for which the integral map K3(A)→K3(FracA)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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.