Some applications of the ultrafilter topology on spaces of valuation domains, Part II

Carmelo Antonio Finocchiaro; Marco Fontana

doi:10.5802/acirm.45

Carmelo Antonio Finocchiaro ¹; Marco Fontana ²

¹ C.A.F. - Dipartimento di Matematica Università degli studi Roma Tre Largo San Leonardo Murialdo 1, 00146 Roma, Italy
² M.F. - Dipartimento di Matematica Università degli studi Roma Tre Largo San Leonardo Murialdo 1, 00146 Roma, Italy

Actes des rencontres du CIRM, Volume 2 (2010) no. 2, pp. 111-114.

Abstract

Let $K$ be a field and $A$ be a subring of $K$ . In the present note, we present the main applications of the so called ultrafilter topology on the space $Zar (K | A)$ , introduced in the previous Part I. After recalling that $Zar (K | A)$ is a spectral space, we give an explicit description of $Zar (K | A)$ as the prime spectrum of a ring (even in the case when the quotient field of $A$ is a proper subfield of $K$ ). Moreover, we provide applications of the topological material previously introduced to the study of representations of integrally closed domains and valuative semistar operations.

Published online: 2011-10-19

Zbl: 1439.13014

DOI: 10.5802/acirm.45

Keywords: Valuation domain, (semi)star operation, prime spectrum, Zariski topology, constructible topology, filter and ultrafilter, Prüfer domain.

Author's affiliations:

Carmelo Antonio Finocchiaro ¹; Marco Fontana ²

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Carmelo Antonio Finocchiaro; Marco Fontana. Some applications of the ultrafilter topology on spaces of valuation domains, Part II. Actes des rencontres du CIRM, Volume 2 (2010) no. 2, pp. 111-114. doi : 10.5802/acirm.45. https://pmb.cedram.org/articles/10.5802/acirm.45/

References
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[2] C. A. Finocchiaro, M. Fontana, Some applications of the ultrafilter topology on spaces of valuation domains, Part I, this volume. | DOI

[3] C. A. Finocchiaro, M. Fontana, K. A. Loper, Ultrafilter and constructible topologies on spaces of valuation domains, submitted. | DOI | MR | Zbl

[4] M. Fontana, K. A. Loper, Cancellation properties in ideal systems: a classification of e.a.b. semistar operations, J. Pure Appl. Algebra 213 (2009), no. 11, 2095–2103. | DOI | MR | Zbl

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