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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Nakai conjecture 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. This is the conjectural converse to a result of Alexander Grothendieck. It is known to be true for algebraic curves. The conjecture was proposed by the Japanese mathematician Yoshikazu Nakai. A consequence would be the Zariski-Lipman conjecture, for a complex affine variety V with coordinate ring R: if the derivations of R are a free module over R, then V is smooth.