Real holomorphy rings and the complete real spectrum

— The complete real spectrum of a commutative ring A with 1 is introduced. Points of the complete real spectrum SpercA 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. SpercA is shown to have the structure of a spectral space in the sense of Hochster [5]. The specialization relation on SpercA 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 SpercA, 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 SpercA a une structure d’espace spectral au sens de Hochster [5]. On considère aussi la relation de spécialisation sur SpercA. Nous nous intéressons particulièrement au cas où l’anneau A est un anneau d’holomorphie réel. (∗) This work was initiated in 2003 and completed in 2007. Funding was provided by the bilateral contracts for cooperation between Université Paris 6 and the University of Saskatchewan. The research of the second author was supported also by the Natural Sciences and Engineering Research Council of Canada. (1) Institut de Mathématiques de Jussieu, Université Paris 6, 4 Place Jussieu, 75252 Paris Cedex 05, France [email protected] (2) Department of Mathematics & Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK Canada, S7N 5E6 [email protected]