A nullstellensatz with nilpotents and Zariski’s main lemma on holomorphic functions

The classical Nullstellensatz asserts that a reduced affine variety is known by its closed points; algebraically, a prime ideal in an affine ring is the intersection of the maximal ideals containing it. A leading special case of our theorem says that any affine scheme can be distinguished from its subschemes by its closed points with a bounded index of nilpotency; algebraically, an ideal I in an affine ring A may be written as I = (I (me + I), (*I n*ev-where JV is the set of maximal ideals containing I, and e is an integer depending on the degree of nilpotency of A/I. Our theorem might also be thought of as a sharpening of Zariski’s Main Lemma on holomorphic functions [4]. Roughly speaking, this lemma asserts that if a regular function f on an irreducible affine variety V vanishes to order e at each of a dense set JV of closed points of V, then it vanishes to order e at the generic point; that is, if P is the prime ideal in k[x,,..., x,J defining V, then the eth symbolic power of P, where the intersection is taken over a dense set of maximal ideals #Z of K[x,,..., x,J containing P. Of course this implies that, if I is a P-primary ideal containing Pte), then (*), above, is a sharpening that includes (**). Our proof is related to Zariski’s but is simpler than his. * Both authors are grateful for the support of the NSF during the preparation of thi