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Zariski–Lipman conjecture

Let kk be a field of characteristic zero and let RR be a finitely generated kk-algebra. For a commutative kk-algebra AA, write

Der⁡k(A)={ D∈End⁡k(A)  :  D(ab)=a D(b)+b D(a)  for all a,b∈A },\operatorname{Der}_{k}(A)=\{\, D\in\operatorname{End}_{k}(A) \;:\; D(ab)=a\,D(b)+b\,D(a)\ \text{ for all } a,b\in A \,\},

regarded as an AA-module via (aD)(b)=a D(b)(aD)(b)=a\,D(b).

Then for every prime ideal p⊂R\mathfrak p\subset R, if Der⁡k(Rp)\operatorname{Der}_{k}(R_{\mathfrak p}) is a free RpR_{\mathfrak p}-module, then the local ring RpR_{\mathfrak p} is regular.

Equivalently, in the localized form: for every prime p⊂R\mathfrak p\subset R,

Der⁡k(R)p free over Rp⟹Rp regular,\operatorname{Der}_{k}(R)_{\mathfrak p}\ \text{free over}\ R_{\mathfrak p}\quad\Longrightarrow\quad R_{\mathfrak p}\ \text{regular},

where Der⁡k(R)p\operatorname{Der}_{k}(R)_{\mathfrak p} denotes the localization at p\mathfrak p of the RR-module Der⁡k(R)\operatorname{Der}_{k}(R) (which is canonically isomorphic to Der⁡k(Rp)\operatorname{Der}_{k}(R_{\mathfrak p}), since Der⁡k(R)=Hom⁡R(ΩR/k,R)\operatorname{Der}_{k}(R)=\operatorname{Hom}_{R}(\Omega_{R/k},R) with ΩR/k\Omega_{R/k} a finitely presented RR-module).

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Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Zariski–Lipman conjecture

    In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961. It states that if V is a complex algebraic variety, such that its ring of differential operators is generated by the derivations it contains, then V is a smooth variety. The converse statement, that smooth algebraic varieties have rings of differential operators that are generated by their derivations, is a result of Alexander Grothendieck.

    source: Wikipedia

  2. Zariski–Lipman conjecture on projective derivations and regularity

    Let kk be a field of characteristic zero, and let AA be the affine ring or the local analytic ring of a variety over kk. Zariski–Lipman conjecture. If

    Der1(A)Der^1(A)

    is AA-projective, then AA is regular. This conjecture is described in the source as a long-standing open problem in general cases, with recent progress cited there.

    source: Rui Li, Zida Xiao and Huaiqing Zuo, “The Nakai Conjecture for isolated hypersurface singularities of modality 2”, arXiv:2502.04672 (2025).

Sources & referencesView supporting material

Primary source

Encyclopedia of Mathematics

Additional references

  1. J. Lipman, "Free derivation modules on algebraic varieties," American Journal of Mathematics 87 (1965), 874–88.
  2. Encyclopedia of Mathematics, Zariski–Lipman conjecture.
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