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 ringR. This conjecture states that if the derivations of R are a
free module over R, then V is smooth.[5]
References
^Nakai, Yoshikazu (1961), "On the theory of differentials in commutative rings", Journal of the Mathematical Society of Japan, 13: 63–84,
doi:10.2969/jmsj/01310063,
MR0125131.
^Becker, Joseph (1977), "Higher derivations and the Zariski-Lipman conjecture", Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), Providence, R. I.:
American Mathematical Society, pp. 3–10,
MR0444654.