Let X = operatornameSpec A, where (A,mathfrakm_A,L) is a regular local ring. For each n ≥ 0, let G_n(X) be the Gersten complex associated with the codimensional filtration, whose terms are built from the groups K_n-p(k(x)) at points of codimension p. Gersten's Conjecture. The complex G_n(X) resolves K_n(X), meaning H^p(G_n(X)) = K_n(X) & p = 0 0 & p > 0. This is the acyclicity conjecture for the Gersten complex of a regular local ring and is a central problem in algebraic K-theory. The source does not specify a resolution status for the general statement.

Gersten's acyclicity conjecture for regular local rings

At least 8 years old · documented by 2017

Let $X = \operatorname{Spec} A$ , where $(A,\mathfrak{m}_A,L)$ is a regular local ring. For each $n \geq 0$ , let $\mathcal{G}_n(X)$ be the Gersten complex associated with the codimensional filtration, whose terms are built from the groups $K_{n-p}(k(x))$ at points of codimension $p$ . Gersten's Conjecture. The complex $\mathcal{G}_n(X)$ resolves $K_n(X)$ , meaning

H^p(\mathcal{G}_n(X)) = \begin{cases} K_n(X) & p = 0 \\ 0 & p > 0. \end{cases}

This is the acyclicity conjecture for the Gersten complex of a regular local ring and is a central problem in algebraic $K$ -theory. The source does not specify a resolution status for the general statement.

References

Primary source

C. Skalit, “Regular Morphisms and Gersten's Conjecture”, arXiv:1710.00303 (2017).

algebraic-k-theory gersten-conjecture

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RemarkAI-assistedClaimed by OpenAI. The manuscript claims a two-dimensional ramified regular local ring A of mixed characteristic (0,5) for which K_5(A) to K_5(Frac(A)) has nonzero kernel. This obstructs the natural first injectivity required by the page’s integral Gersten resolution and gives a claimed mixed-characteristic counterexample. It does not concern only Milnor K-theory, rational coefficients, or regular local rings containing a field.See full solutionHide full solution

Claimed by OpenAI. The manuscript claims a two-dimensional ramified regular local ring A of mixed characteristic (0,5) for which K_5(A) to K_5(Frac(A)) has nonzero kernel. This obstructs the natural first injectivity required by the page’s integral Gersten resolution and gives a claimed mixed-characteristic counterexample. It does not concern only Milnor K-theory, rational coefficients, or regular local rings containing a field.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026/An-Integral-Counterexample-to-Gerstens-Conjecture-September-25-2026.pdf

OpenAI-209-01-An-integral-counterexample-to-Gersten-s-conjecture.pdf469,239 bytesOpen

RemarkAI-assistedClaimed by OpenAI. The manuscript claims a two-dimensional ramified regular local ring A of mixed characteristic (0,5) with nonzero kernel in the integral map K_3(A) to K_3(Frac(A)). This contradicts the natural first injectivity needed for the page’s Gersten resolution in degree three if the claim holds. The statement is a ramified mixed-characteristic example, not a claim about rational coefficients or only Milnor K-theory.See full solutionHide full solution

Claimed by OpenAI. The manuscript claims a two-dimensional ramified regular local ring A of mixed characteristic (0,5) with nonzero kernel in the integral map K_3(A) to K_3(Frac(A)). This contradicts the natural first injectivity needed for the page’s Gersten resolution in degree three if the claim holds. The statement is a ramified mixed-characteristic example, not a claim about rational coefficients or only Milnor K-theory.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Integral-Degree-Three-Gersten-Counterexample-September-26-2026/paper.pdf

OpenAI-209-02-An-integral-degree-three-Gersten-counterexample.pdf341,624 bytesOpen