Semistar-Krull and Valuative Dimension of Integral Domains

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{-}$\dim(D[X])\leq 2n+1$\mathrm{.}Wealsoestablishthatif$D$isa$\star$-NoetheriandomainorisaPr\ddot{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 $\star [X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n]$-${\dim }_v(D[X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n])=2n$, and that if $\star$-${\dim }_v(D)<\mathrm{\infty }$ 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 Pr"{u}fer $\star$-multiplication domain. We define the $\star$-Jaffard domains as domains $D$ such that $\star$-$\dim (D)<\mathrm{\infty }$ 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 $(\star ,{\star }^{\mathrm{\prime }})$-linked overring $T$ of $D$, is a ${\star }^{\mathrm{\prime }}$-Jaffard domain, where ${\star }^{\mathrm{\prime }}$ 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.