Nakai conjecture
Template:Short description In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961.[1] 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.[2]
The Nakai conjecture is known to be true for algebraic curves[3] and Stanley–Reisner rings.[4] 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.[5]