Real holomorphy rings and the complete real spectrum

Gondard, D.¹ ; Marshall, M.² 

¹ Institut de Mathématiques de Jussieu, Université Paris 6, 4 Place Jussieu, 75252 Paris Cedex 05, France
² Department of Mathematics & Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK Canada, S7N 5E6

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Numéro spécial : Actes de la Conférence sur les anneaux ordonnés en honneur de Melvin Henriksen, Université d’état de Louisiane à Bâton Rouge, 2007, Tome 19 (2010) no. S1, pp. 57-74.

Abstract (VO)

The complete real spectrum of a commutative ring $A$ with $1$ is introduced. Points of the complete real spectrum ${Sper}^{c}A$ are triples $\alpha =(𝔭,v,P)$, where $𝔭$ is a real prime of $A$, $v$ is a real valuation of the field $k\left(𝔭\right):=qf(A/𝔭)$ and $P$ is an ordering of the residue field of $v$. ${Sper}^{c}A$ is shown to have the structure of a spectral space in the sense of Hochster [5]. The specialization relation on ${Sper}^{c}A$ is considered. Special attention is paid to the case where the ring $A$ in question is a real holomorphy ring.

Résumé

Nous introduisons la notion de spectre réel complet d’un anneau $A$ commutatif avec unité. Les points de ce spectre réel complet, noté ${Sper}^{c}A$, sont les triplets $\alpha =(𝔭,v,P)$, où $𝔭$ est un idéal premier de $A$, $v$ une valuation réelle du corps $k\left(𝔭\right):=qf(A/𝔭)$ et $P$ un ordre du corps résiduel de $v$. Nous montrons que ${Sper}^{c}A$ a une structure d’espace spectral au sens de Hochster [5]. On considère aussi la relation de spécialisation sur ${Sper}^{c}A$. Nous nous intéressons particulièrement au cas où l’anneau $A$ est un anneau d’holomorphie réel.

Détail

DOI : 10.5802/afst.1275

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