Gondard, D.; Marshall, M.
Real holomorphy rings and the complete real spectrum
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6 : Tome 19 (2010) no. S1 , p. 57-74
MR 2675721 | Zbl 1209.13026
doi : 10.5802/afst.1275
URL stable : http://www.numdam.org/item?id=AFST_2010_6_19_S1_57_0

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.
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.

Bibliographie

[1] C. Andradas, L. Bröcker, J. Ruiz, Constructible sets in real geometry, Springer 1996 MR 1393194 | Zbl 0873.14044

[2] E. Becker, D. Gondard, On the space of real places of a formally real field, Real analytic and algebraic geometry, Walter de Gruyter (1995), 21–46 MR 1320309 | Zbl 0869.12002

[3] E. Becker, V. Powers, Sums of powers in rings and the real holomorphy ring, J. reine angew. Math. 480 (1996), 71–103 MR 1420558 | Zbl 0922.12003

[4] J. Bochnak, M. Coste, M.-F. Roy, Géométrie algébrique réelle, Ergeb. Math. Springer 1987 MR 949442 | Zbl 0633.14016

[5] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142 (1969), 43–60 MR 251026 | Zbl 0184.29401

[6] R. Huber, Bewertungsspektrum und rigide geometrie, Regensburger Math. Schriften 23 1993 MR 1255978 | Zbl 0806.13001

[7] R. Huber, M. Knebusch, On valuation spectra, Contemporary Math. 155 (1994), 167–206 MR 1260707 | Zbl 0799.13002

[8] M. Knebusch, D. Zhang, Manis valuations and Prüfer extensions I, Springer 2002 MR 1937245 | Zbl 1033.13001

[9] T.-Y. Lam, An introduction to real algebra, Rky. Mtn. J. Math. 14 (1984), 767–814 MR 773114 | Zbl 0577.14016

[10] M. Marshall, Spaces of orderings and abstract real spectra, Lecture Notes in Mathematics 1636, Springer 1996 MR 1438785 | Zbl 0866.12001

[11] M. Marshall, A real holomorphy ring without the Schmüdgen property, Canad. Math. Bull. 42 (1999), 354–358 MR 1703695 | Zbl 0971.12002

[12] M. Marshall, Real reduced multirings and multifields J. Pure and Applied Algebra 205 (2006), 452–468 MR 2203627 | Zbl 1089.14009

[13] M.J. de la Puente, Riemann surfaces of a ring and compactifications of semi-algebraic sets, Doctoral Dissertation, Stanford 1988

[14] M.J. de la Puente, Specializations and a local homomorphism theorem for real Riemann surfaces of rings, Pac. J. Math. 176 (1996), 427–442 MR 1435000 | Zbl 0868.13004

[15] K. Schmüdgen, The K–moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), 203–206 MR 1092173 | Zbl 0744.44008

[16] H. Schülting, On real places of a field and the real holomorphy ring, Comm. Alg. 10 (1982), 1239–1284 MR 660344 | Zbl 0509.14026

[17] M. Schweighofer, Iterated rings of bounded elements and generalizations of Schmüdgen’s Positivstellensatz, J. reine angew. Math. 554 (2003), 19–45 MR 1952167 | Zbl 1096.13032