Semistar-Krull and Valuative Dimension of Integral Domains

TL;DR

This paper introduces new concepts of semistar dimension and valuative dimension for integral domains, establishing their properties, relationships, and applications to classes like Jaffard and Krull domains.

Contribution

It defines the semistar dimension and valuative dimension, explores their behavior under polynomial extensions, and characterizes certain domains like Jaffard and Krull domains using these dimensions.

Findings

Semistar dimension of D increases by 1 in polynomial ring D[X] for specific classes.

Semistar valuative dimension doubles in polynomial extensions under certain conditions.

Krull domains are characterized as w_D-Jaffard domains.

Abstract

Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $\star[X]$ on the polynomial ring $D[X]$, such that, if $n:=\star$-$\dim(D)$, then $n+1\leq \star[X]\text{-}\dim(D[X])\leq 2n+1$. We also establish that if $D$ is a $\star$-Noetherian domain or is a Pr\"{u}fer $\star$-multiplication domain, then $\star[X]\text{-}\dim(D[X])=\star\text{-}\dim(D)+1$. Moreover we define the semistar valuative dimension of the domain $D$, denoted by $\star$-$\dim_v(D)$, to be the maximal rank of the $\star$-valuation overrings of $D$. We show that $\star$-$\dim_v(D)=n$ if and only if $\star[X_1,...,X_n]$-$\dim_v(D[X_1,...,X_n])=2n$, and that if $\star$-$\dim_v(D)<\infty$ then $\star[X]$-$\dim_v(D[X])=\star$-$\dim_v(D)+1$. In general $\star$-$\dim(D)\leq\star$-$\dim_v(D)$ and equality holds if $D$ is a $\star$-Noetherian domain or is a Pr\"{u}fer $\star$-multiplication domain. We define the $\star$-Jaffard domains as domains $D$ such that $\star$-$\dim(D)<\infty$ and $\star$-$\dim(D)=\star$-$\dim_v(D)$. As an application, $\star$-quasi-Pr\"{u}fer domains are characterized as domains $D$ such that each $(\star,\star')$-linked overring $T$ of $D$, is a $\star'$-Jaffard domain, where $\star'$ 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.