A Property of Ideals in Polynomial Rings

Every ideal in the polynomial ring in n variables over an infinite field has a reduction generated by n elements. Eisenbud and Evans [2] proved that every ideal in k[Xx,...,Xn] can be generated up to radical by n elements (where k is a field). Avinash Sathaye [7] and Mohan Kumar [5] proved a locally complete intersection in k[ Xv ..., Xn] can be generated by n elements. In this short note we show that every ideal in k[Xx,..., Xn] has a nice approximation generated by n elements. More precisely, we prove the following. Theorem. Let k be an infinite field. Then every ideal I in k[Xx,..., Xn] has a reduction J generated by n elements. By [6], J is a reduction of / if there exists an integer r such that JV = Ir+1. Northcott and Rees [6] point out that / can be regarded as a simplified version of / preserving many properties of /, in particular the multiplicities at minimal prime over-ideals. Moreover, / has the same radical as / and if / is locally a complete intersection, then it is the only reduction of itself, hence a connection between our result and those of Eisenbud and Evans and Sathaye and Kumar. Proof of the theorem. Since k[Xv..., Xn] is a UFD, we can assume that n > drm(A/I) + 2. Let gv...,gr be a system of generators of I. Set A = k[Xv..., Xn] and B = k[gu..., gr] c A. The dimension of B is at most n, since its quotient field is a subfield of k(Xv..., X„) and therefore has transcendence degree < n. Denote by P the ideal of B generated by g,,..., gr. Let P be the image of F in BP, the localization of B at P (i.e. P is the maximal ideal of BP). Since BP is local of dimension < n, by Burch [1] P has a reduction Q generated by n elements hv...,h„. Let r be an integer such that QPr = Pr+1. Since Q is F-primary, there exists a unique F-primary ideal Q c B, the image of which in BP coincides with Q. Since F is a maximal ideal of B, we see that the ideals QPr and Pr+1 are F-primary and their images in BP coincide. Therefore, QPr = Pr+1 in B, since there is a one-to-one correspondence between the F-primary ideals of BP and the F-primary ideals of B. Let J be the extension of Q in A. Since the extension of F is / and QPr = Pr+l, we see that JIr = Ir+1, i.e. / is a reduction of I. Received by the editors September 13, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 13F20, 13A15. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 399 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 400 GENNADY LYUBEZNIK Let s g B\P be an element such that Qs = (hx,..., hn). Thus Js = (hx,..., hn). Since Q is F-primary and F is a maximal ideal of B, we see that (s) + Q = (1), i.e., regarding 5 as an element of A, that (s) + J = (1). The conditions (s) + J = (1) and Js = (hx,..., hn) imply that J/J2 is generated by n elements hx,...,h„. Now Theorem 5 of Mohan Kumar [4] tells us that J is generated by n elements, since n > dim(A/J) + 2. Q.E.D. Remarks. 1. In a similar way one can prove that every ideal in a finitely generated «-dimensional algebra over an infinite field has a reduction generated by n + 1 elements. 2. For every prime ideal F c B containing / we have the dimension of {J + P ■ I)/P ■ I as a vector space over the quotient field of A/P is > height F This improves Eisenbud and Evans [2, 3] who proved only that J <t P ■ I. 3. Some generalization of our theorem is possible to the case / c A[X], where A is an (n — l)-dimensional finitely generated algebra over an infinite field. For this one has to use S. Mandall's extension of Mohan Kumar's theorem to ideals in A [X] (cf. [5]). Conjecture. Let A be a commutative Noetherian ring of dimension n — 1 such that the residue field of every maximal ideal of A is infinite. Let / be an ideal of A ox A\X\. Then I has a reduction generated by n elements.