Semistar-Krull and valuative dimension of integral domains

Abstract

Given a stable semistar operation of finite type ⋆ on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite type ⋆[X] on the polynomial ring D[X], such that, if n := ⋆-dim(D), then n+1 ≤ ⋆[X]-dim(D[X]) ≤ 2n+1. We also establish that if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain, then ⋆[X]-dim(D[X]) = ⋆- dim(D)+1. Moreover we define the semistar valuative dimension of the domain D, denoted by ⋆-dim v (D), to be the maximal rank of the ⋆-valuation overrings of D. We show that ⋆-dim v (D) = n if and only if ⋆[X 1, . . . , X n ]-dim(D[X 1, . . . , X n ]) = 2n, and that if ⋆-dim v (D) < ∞ then ⋆[X]-dim v (D[X]) = ⋆-dim v (D) + 1. In general ⋆-dim(D) ≤ ⋆-dim v (D) and equality holds if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain. We define the ⋆-Jaffard domains as domains D such that ⋆-dim(D) < ∞ and ⋆-dim(D) = ⋆-dim v (D). As an application, ⋆-quasi-Prüfer domains are characterized as domains D such that each (⋆, ⋆′)-linked overring T of D, is a ⋆′-Jaffard domain, where ⋆′ is a stable semistar operation of finite type on T. As a consequence of this result we obtain that a Krull domain D, must be a w D -Jaffard domain.