Nakai conjecture

{{short description|Conjecture in algebraic geometry}}

In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961.{{citation

| last = Nakai | first = Yoshikazu

| journal = Journal of the Mathematical Society of Japan

| mr = 0125131

| pages = 63–84

| title = On the theory of differentials in commutative rings

| volume = 13

| year = 1961

| doi=10.2969/jmsj/01310063| doi-access = free

}}.

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.{{citation

| last = Schreiner | first = Achim

| doi = 10.1007/BF01193737

| issue = 6

| journal = Archiv der Mathematik

| mr = 1274105

| pages = 506–512

| title = On a conjecture of Nakai

| volume = 62

| year = 1994}}. Schreiner cites this converse to EGA 16.11.2.

The Nakai conjecture is known to be true for algebraic curves{{citation

| last1 = Mount | first1 = Kenneth R.

| last2 = Villamayor | first2 = O. E.

| journal = Osaka Journal of Mathematics

| mr = 0327731

| pages = 325–327

| title = On a conjecture of Y. Nakai

| volume = 10

| year = 1973}}. and Stanley–Reisner rings.{{citation

| last = Schreiner | first = Achim

| doi = 10.1007/BF01193737

| issue = 6

| journal = Archiv der Mathematik

| mr = 1274105

| pages = 506–512

| title = On a conjecture of Nakai

| volume = 62

| year = 1994}}. 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.{{citation

| last = Becker | first = Joseph

| contribution = Higher derivations and the Zariski-Lipman conjecture

| location = Providence, R. I.

| mr = 0444654

| pages = 3–10

| publisher = American Mathematical Society

| title = Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975)

| year = 1977}}.