Nakai conjecture

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]

References

1. 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, MR 0125131.
2. Schreiner, Achim (1994), "On a conjecture of Nakai", Archiv der Mathematik, 62 (6): 506–512, doi:10.1007/BF01193737, MR 1274105. Schreiner cites this converse to EGA 16.11.2.
3. Mount, Kenneth R.; Villamayor, O. E. (1973), "On a conjecture of Y. Nakai", Osaka Journal of Mathematics, 10: 325–327, MR 0327731.
4. Schreiner, Achim (1994), "On a conjecture of Nakai", Archiv der Mathematik, 62 (6): 506–512, doi:10.1007/BF01193737, MR 1274105.
5. 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, MR 0444654.