Semistar-Krull and Valuative Dimension of Integral 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 $n+1\le \star [X]\text{-}\dim (D[X])\le 2n+1$. We also establish that if $D$ is a $\star$-Noetherian domain or is a Prüfer $\star$-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 $\star$-$\dim (D)\le \star$-${\dim }_v(D)$ and equality holds if $D$ is a $\star$-Noetherian domain or is a Prüfer $\star$-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-Prüfer domains are characterized as domains $D$ such that each $(\star ,{\star }^{\prime })$-linked overring $T$ of $D$, is a ${\star }^{\prime }$-Jaffard domain, where ${\star }^{\prime }$ 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.

Final version: Remark 2.2 change to Ptoposition 2.2 and added Example 4.4

Cited by in corpus (1)

• Universally catenarian integral domains, strong S-domains and semistar operations