GOING-DOWN AND SEMISTAR OPERATIONS

If D ⊆ T is an extension of (commutative integral) domains and ⋆ (resp., ⋆′) is a semistar operation on D (resp., T), we define what it means for D ⊆ T to satisfy the (⋆,⋆′)-GD property. Sufficient conditions are given for (⋆,⋆′)-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 ⋆ is a semistar operation on a domain D, we define what it means for D to be a ⋆-GD domain, generalizing the notion of a going-down domain. In determining whether a domain D is a $\widetilde{\star}\mbox{-GD}$ domain, the domain extensions T of D for which $(\widetilde{\star},\star')\mbox{-GD}$ is tested can be the $\widetilde{\star}$-valuation overrings of D, the simple overrings of D, or all T. P ⋆ MDs are characterized as the $\widetilde{\star}$-treed (resp., $\widetilde{\star}\mbox{-GD}$) domains D which are $\widetilde{\star}$-finite conductor domains such that $D^{\widetilde{\star}}$ is integrally clos...