Elementary properties of minimal and maximal points in Zariski spectra

Abstract

We investigate connections between arithmetic properties of rings and topological properties of their prime spectrum. Any property that the prime spectrum of a ring may or may not have, defines the class of rings whose prime spectrum has the given property. We ask whether a class of rings defined in this way is axiomatizable in the model theoretic sense. Answers are provided for a variety of different properties of prime spectra, e.g., normality or complete normality, Hausdorffness of the space of maximal points, compactness of the space of minimal points.

Graph components of prime spectra

“…The subspace Min( X) ⊆ X is always Hausdorff, but need not be compact. A study of the spaces Max(Spec( A)) and Min(Spec( A)) (where A is a ring) is contained in [22].…”

“…Let X be a normal spectral space, i.e., every point of X specializes to a unique maximal point, say x → μ(x) ∈ Max( X), [3], [22,Section 4]. Normal spectral spaces abound in real algebra: Real spectra of rings, [2,Chapter 7], are normal spectral spaces.…”

“…Various different characterizations are given in [3] and in [22,Sections 4,5]. In the present context the most useful characterization is: A spectral space is normal if every point specializes to only one maximal point.…”

“…If a spectral space is normal (cf. [3], [22,Section 4]) then its space of maximal points is compact and is homeomorphic to the space of graph components, 4.7. The space of graph components is Boolean if and only if the topological components are graph connected, 4.9.…”

Convex subrings of partially ordered rings

“…The real spectrum of a poring is described in [20], Section 4, and in [22]. The notation and basic facts concerning (prime) spectra can be looked up in [24].…”