Nakai conjecture

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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]

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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 .