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Math-2211 · mathematics · Algebra

Zariski–Lipman conjecture

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $V$ be a complex algebraic variety with coordinate ring $R$. If the module of derivations of $R$ is free over $R$, must $V$ be smooth?

Formulation references: [S1], [S2].

Short English statement

Let $V$ be a complex algebraic variety with coordinate ring $R$. If the module of derivations of $R$ is free over $R$, must $V$ be smooth?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2211-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2211

Manuscript 1 (family 048, 2026-09-23): The normal integral affine complex surface has Der_C(A) free of rank two but a nonregular maximal localization. Thus its tangent/derivation module is free while the variety is singular, directly refuting the stated implication.

Zariski–Lipman conjecture

OpenAI's claim

A singular normal affine complex surface is claimed to have a globally free rank-two tangent sheaf, contradicting the characteristic-zero Lipman--Zariski assertion. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
OpenAI claims a normal integral $\mathbb C$-algebra $A$, required to be finitely generated and two-dimensional in the Krull sense, together with a maximal ideal $\mathfrak m$ with \[ \operatorname{Der}_{\mathbb C}(A)\simeq A^{\oplus2},\qquad A_{\mathfrak m}\text{ not regular}. \] Thus the associated normal affine complex surface is singular despite a globally free rank-two tangent module. Freeness is as an $A$-module and regularity is the local-ring condition.

Outline of the argument

Manuscript 1 (family 048, 2026-09-23): The analytic seed is a negative-curve neighborhood whose gluing is modified without changing its normal bundle. Two vector fields are built with a nonvanishing determinant off the curve by simultaneous point corrections and global residue constraints; algebraization preserves the completed singularity and allows tangent freeness to descend.

Formal verification

Family 048 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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