GOING-DOWN AND SEMISTAR OPERATIONS

Abstract

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 [Formula: see text] domain, the domain extensions T of D for which [Formula: see text] is tested can be the [Formula: see text]-valuation overrings of D, the simple overrings of D, or all T. P ⋆ MD s are characterized as the [Formula: see text]-treed (resp., [Formula: see text]) domains D which are [Formula: see text]-finite conductor domains such that [Formula: see text] is integrally closed. Several characterizations are given of the [Formula: see text]-Noetherian domains D of [Formula: see text]-dimension 1 in terms of the behavior of the (⋆,⋆′)-linked overrings of D and the ⋆-Nagata rings Na(D,⋆).