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Math-2517 · mathematics · Algebra

Integral K3 injectivity for regular local domains

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every Noetherian regular local domain A with fraction field F, is the canonical map $K_{3}(A)\to K_{3}(F)$ injective on integral algebraic K-theory, including ramified mixed characteristic?

Short English statement

For every Noetherian regular local domain A with fraction field F, is the canonical map $K_{3}(A)\to K_{3}(F)$ injective on integral algebraic K-theory, including ramified mixed characteristic?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2517-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2517

The specified two-dimensional ramified regular local domain A and integral class c have c nonzero (detected after inverting the uniformizer modulo 5) and restriction zero to Frac A. This directly refutes universal integral K_3 injectivity. Within these assumptions, this is a claimed negative answer to the stated question.

Integral K3 injectivity for regular local domains

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. A different two-dimensional ramified ring in mixed characteristic $(0,5)$ is claimed to violate integral Gersten injectivity in degree three.

Mathematical statement of the claimed result

Result 1.
Let $\zeta$ be a primitive fifth root of unity, $V=\mathcal O_{\mathbb Q_5(\zeta)}$, and $\pi=1-\zeta$. In $\mathbb P^2_V$, take the hypersurface $(X-Z)^5-XY^4+Y^5+\zeta Z^5=0$, and localize at its special-fiber point $[0:0:1]$, obtaining $A$. The source claims that $A$ is a regular Noetherian local domain of dimension two and mixed characteristic $(0,5)$, with $\pi\in\mathfrak m_A^2$, and some integral class $c\in K_3(A)$ satisfies \[ c|_{\operatorname{Frac}A}=0,\qquad c|_{A[1/\pi]}\bmod5\ne0\text{ in }G_3(A[1/\pi];\mathbb Z/5). \] Thus the integral restriction $K_3(A)\to K_3(\operatorname{Frac}A)$ has nonzero kernel.

Outline of the argument

A divisor on a cyclotomic quintic supplies a Bott-symbol class whose coefficient boundary vanishes by a Steinberg relation. The resulting integral lift is generically zero by support, and a norm-valuation test detects it using the different residue-degree behavior of closed points.

Formal verification

Family 209 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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