The Lipman-Zariski conjecture in characteristic zero

Theorem 1.1: there are a finitely generated normal integral $\mathbb C$ -algebra $A$ of dimension two and a maximal ideal $\mathfrak m$ with $\mathrm{Der}_{\mathbb C}(A)\cong A^{\oplus2}$ and $A_{\mathfrak m}$ not regular; the surface can be smooth away from that point. By Jorder's theorem no local tangent frame there commutes. The paper also answers negatively, in characteristic zero, the derivation-blowup question of Barajas, Chavez-Martinez and Romano-Velazquez (2026, Question 1). It does not address higher-dimensional non-isolated cases left open by earlier results or give a counterexample with a commuting frame (impossible by Jorder).

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Prior state unknown→disproved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Lipman-Zariski conjecture in characteristic zero · Lipman-Zariski conjecture

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VibeMathed record: The Lipman-Zariski conjecture in characteristic zero

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript was read against the conjecture; it gives a finitely generated normal two-dimensional $\mathbb C$ -algebra $A$ with $\mathrm{Der}_{\mathbb C}(A)\simeq A^{\oplus2}$ globally free and a maximal ideal at which $A$ is not regular, which contradicts the conjecture as posed. The construction starts from an analytic Gorenstein surface singularity (exceptional curve of genus 21, self-intersection $-10$ ) and algebraizes it; the paper checks that the example lies outside the Graf, Bergner-Graf, log canonical and Biswas-Gurjar-Kolte criteria. Not refereed. No Lean formalization in the release.