Going-down and semistar operations

چکیده مقاله

If $D\subseteq T$ is an extension of (commutative integral) domains
and $\star$ (resp., $\star'$) is a semistar operation on $D$ (resp.,
$T$), we define what it means for $D\subseteq T$ to satisfy the
$(\star,\star')$-$\GD$ property. Sufficient conditions are given for
$(\star,\star')$-$\GD$, generalizing classical sufficient conditions
for $\GD$ such as flatness, openness of the contraction map of
spectra and the hypotheses of the classical going-down theorem. If
$\star$ is a semistar operation on a domain $D$, we define what it
means for $D$ to be a $\star$-$\GD$ domain, generalizing the notion
of a going-down domain. In determining whether a domain $D$ is a
$\widetilde{\star}$-$\GD$ domain, the domain extensions $T$ of $D$
for which $(\widetilde{\star},\star')$-$\GD$ is tested can be the
$\widetilde{\star}$-valuation overrings of $D$, the simple overrings
of $D$, or all $T$. $P\star MD$s are characterized as the
$\widetilde{\star}$-treed (resp., $\widetilde{\star}$-$\GD$) domains
$D$ which are $\widetilde{\star}$-finite conductor domains such that
$D^{\widetilde{\star}}$ is integrally closed. Several
characterizations are given of the $\widetilde{\star}$-Noetherian
domains $D$ of $\widetilde{\star}$-dimension $1$ in terms of the
behavior of the $(\star,\star')$-linked overrings of $D$ and the
$\star$-Nagata rings $Na(D,\star)$.