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.
Spirito, D. (2017). Non-compact subsets of the Zariski space of an integral domain. ILLINOIS JOURNAL OF MATHEMATICS, 60(3-4), 791-809.
Non-compact subsets of the Zariski space of an integral domain
Spirito, Dario
2017-01-01
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.File in questo prodotto:
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