Nakai 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.
The Nakai conjecture is known to be true for algebraic curves and Stanley–Reisner rings. A proof of the conjecture would also establish the Zariski–Lipman conjecture, for a complex variety "V" with coordinate ring "R". This conjecture states that if the derivations of "R" are a free module over "R", then "V" is smooth.