Non-compact subsets of the Zariski space of an integral domain

2017/05/03 by Spirito, Dario
#13A15 #13A18 #13B22 #13F30 (Primary) #54D30 (Secondary) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.01301

Abstract

Let V be a minimal valuation overring of an integral domain D and let Zar(D) be the Zariski space of the valuation overrings of D. Starting from a result in the theory of semistar operations, we prove a criterion under which the set Zar(D)∖\V\ is not compact. We then use it to prove that, in many cases, Zar(D) is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings.

Related

• Non-integrally closed Kronecker function rings and integral domains with a unique minimal overring
• Isolated points of the Zariski space
• The constructible topology on spaces of valuation domains
• A connectedness theorem for spaces of valuation rings
• Affine schemes and topological closures in the Zariski-Riemann space of valuation rings