On the Dimension Theory of Polynomial Rings over Pullbacks

Since Seidenberg’s (1953-54) papers [35, 36] and Jaffard’s (1960) pamphlet [28] on the dimension theory of commutative rings, the literature abounds in works exploring the prime ideal structure of polynomial rings, including four pioneering articles by Arnold and Gilmer on dimension sequences [3, 4, 5, 6]. Of particular interest is Bastida-Gilmer’s (1973) precursory article [8] which established a formula for the Krull dimension of a polynomial ring over a D+M issued from a valuation domain. During the last three decades, numerous papers provided in-depth treatments of dimension theory and other related notions (such as the S-property, strong S-property, and catenarity) in polynomial rings over various pullback constructions. All rings considered in this paper are assumed to be integral domains. A polynomial ring over an arbitrary domain R is subject to Seidenberg’s inequalities: n + dim(R) ≤ dim(R[X1, ..., Xn]) ≤ n + (n + 1) dim(R), ∀ n ≥ 1. A finite-dimensional domain R is said to be Jaffard if dim(R[X1, ..., Xn]) = n+dim(R) for all n ≥ 1; equivalently, if dim(R) = dimv(R), where dim(R) denotes the Krull dimension of R and dimv(R) its valuative dimension (i.e., the supremum of dimensions of the valuation overrings of R). The study of this class was initiated by Jaffard [28]. For the convenience of the reader, recall that, in general, for a domain R with dimv(R) < ∞ we have: dim(R) ≤ dimv(R), dimv(R[X1, ..., Xn]) = n + dimv(R) for all n ≥ 1, and dim(R[X1, ..., Xn]) = n + dimv(R) for all n ≥ dimv(R) − 1 (Cf. [2, 11, 18, 26, 28]). As the Jaffard property does not carry over to localizations (see Example 3.5 below), R is said to be locally Jaffard if Rp is a Jaffard domain for each prime ideal p of R; equivalently, SR is a Jaffard domain for each multiplicative subset S of R. A locally Jaffard domain is Jaffard [2]. The class of (locally) Jaffard domains contains most classes involved in dimension theory, including Noetherian domains [31], Prüfer domains [26], and universally catenarian domains [10]. In order to treat Noetherian domains and Prüfer domains in a unified manner, Kaplansky [31] introduced the following concepts: A domain R is called an Sdomain if, for each height-one prime ideal p of R, the extension pR[X ] in R[X ] has height 1 too; and R is said to be a strong S-domain if R p is an S-domain for each prime ideal p of R. A strong S-domain R satisfies dim(R[X ]) = dim(R)+1. Notice that while R[X ] is always an S-domain for any domain R [24], R[X ] need not be a strong S-domain even when R is a strong S-domain [12]. Thus R is called a stably strong S-domain (also called a universally strong S-domain) if the polynomial ring R[X1, ..., Xn] is a strong S-domain for each positive integer n. A stably strong S-domain is locally Jaffard [2, 29, 32].