Chains of prime ideals in polynomial rings

Abstract

Let R be a domain and R[X] the ring of polynomials in one variable over R; let P be a prime ideal of R whose height is equal to n. Let ,P be a prime ideal of R [X] such that .Y n R = P, 9 # P[X]. The objective of this paper is to study the saturated chains of prime ideals between (0) and .Y and between (0) and P[X]. By classical results of Seidenberg [ 1 I] and Jaffard [6] it is already known that n + 1 < height 9 Q 2n + 1 and that height 9 = height P[X] + 1. In Section 2, we show that the set {r/./r is the length of some saturated chain of prime ideals in R[X] between (0) and 9’) is independent of the choice of 9’. In the special case of R Noetherian, this has already been proved by Houston and McAdam [5]. In Section 3, we show that for every integer t such that n + 1 < t < height ,P, there exists a saturated chain of prime ideals in R[X] between (0) and _P whose length is equal to t; similarly, we show that for every integer u such that n + 1 < u < height P[X], there exists a saturated chain of prime ideals in R[X] between (0) and P[X] whose length is equal to U. Furthermore, we show that such chains can be chosen such that the chains of the intersections with R are also saturated. In Section 4, we show that there is no rule at all that governs the existence of saturated chain of prime ideals in R[X] between (0) and 9 whose length are less than or equal to n. More precisely, we show that given two positive integers n and m such that n + 1 Q m < 2n + 1, and given integers U, ,..., us such that 2 < u1 < ... < u, < n, there exists a domain R and a prime ideal P of R with height equal to n such that, for every prime ideal 9 of R [X] such * Part of this research was done while the second author was visiting Columbia University in the City of New York, supported by the Conselho National de Desenvolvimento Cientifico e Tecnolbgico do Brasil.