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.

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

Spirito D.
2016

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11390/1215698
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