Real holomorphy rings and the complete real spectrum

D. Gondard; M. Marshall

doi:10.5802/afst.1275

D. Gondard ¹; M. Marshall 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, Serie 6, Volume 19 (2010) no. S1, pp. 57-74.

Abstract
Abstract

The complete real spectrum of a commutative ring $A$ with $1$ is introduced. Points of the complete real spectrum ${Sper}^{c} A$ are triples $α = (𝔭, v, P)$ , where $𝔭$ is a real prime of $A$ , $v$ is a real valuation of the field $k (𝔭) : = qf (A / 𝔭)$ 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.

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 $α = (𝔭, v, P)$ , où $𝔭$ est un idéal premier de $A$ , $v$ une valuation réelle du corps $k (𝔭) : = qf (A / 𝔭)$ 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.

MR Zbl

DOI: 10.5802/afst.1275

Author's affiliations:

D. Gondard ¹; M. Marshall 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

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     title = {Real holomorphy rings and the complete real spectrum},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {57--74},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 19},
     number = {S1},
     year = {2010},
     doi = {10.5802/afst.1275},
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D. Gondard; M. Marshall. Real holomorphy rings and the complete real spectrum. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 19 (2010) no. S1, pp. 57-74. doi : 10.5802/afst.1275. https://afst.centre-mersenne.org/articles/10.5802/afst.1275/

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