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Bulletin of the American Mathematical Society

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On the real spectrum of a ring and its application to semialgebraic geometry


Author: Eberhard Becker
Journal: Bull. Amer. Math. Soc. 15 (1986), 19-60
MSC (1985): Primary 14G30, 12D15
MathSciNet review: 838786
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  • [A] E. Artin, Über die Zerlegung definiter Funktionen in Quadrate, Hamb. Abh. 5 (1927), 100-115.
  • Eberhard Becker, Valuations and real places in the theory of formally real fields, Real algebraic geometry and quadratic forms (Rennes, 1981) Lecture Notes in Math., vol. 959, Springer, Berlin-New York, 1982, pp. 1–40. MR 683127
  • [Bi] G. Birkhoff, Lattice theory, Amer. Math. Soc., Providence, R.I., 1967.
  • [Bo1] N. Bourbaki, Topologie générale, Chapter 1, Hermann, Paris, 1951.
  • N. Bourbaki, Éléments de mathématique. Fascicule XXVIII. Algèbre commutative. Chapitre 3: Graduations, filtra- tions et topologies. Chapitre 4: Idéaux premiers associés et décomposition primaire, Actualités Scientifiques et Industrielles, No. 1293, Hermann, Paris, 1961 (French). MR 0171800
  • Ludwig Bröcker, Positivbereiche in kommutativen Ringen, Abh. Math. Sem. Univ. Hamburg 52 (1982), 170–178 (German). MR 710541, 10.1007/BF02941874
  • Ludwig Bröker, Real spectra and distributions of signatures, Real algebraic geometry and quadratic forms (Rennes, 1981) Lecture Notes in Math., vol. 959, Springer, Berlin-New York, 1982, pp. 249–272. MR 683137
  • Ludwig Bröcker, Minimale Erzeugung von Positivbereichen, Geom. Dedicata 16 (1984), no. 3, 335–350 (German). MR 765338, 10.1007/BF00147875
  • Ludwig Bröcker, Spaces of orderings and semialgebraic sets, Quadratic and Hermitian forms (Hamilton, Ont., 1983) CMS Conf. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 1984, pp. 231–248. MR 776457
  • [Bru1] G. W. Brumfiel, Partially ordered rings and semialgebraic geometry, Cambridge, 1979.
  • G. W. Brumfiel, Witt rings and 𝐾-theory, Rocky Mountain J. Math. 14 (1984), no. 4, 733–765. Ordered fields and real algebraic geometry (Boulder, Colo., 1983). MR 773113, 10.1216/RMJ-1984-14-4-733
  • Ludwig Bröcker and Johann-Heinrich Schinke, On the 𝐿-adic spectrum, Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie [Series of the Mathematical Institute of the University of Münster, Series 2], vol. 40, Universität Münster, Mathematisches Institut, Münster, 1986. MR 859868
  • Michel Coste, Ensembles semi-algébriques, Real algebraic geometry and quadratic forms (Rennes, 1981) Lecture Notes in Math., vol. 959, Springer, Berlin-New York, 1982, pp. 109–138 (French). MR 683131
  • Marie-Françoise Coste and Michel Coste, Topologies for real algebraic geometry, Topos theoretic methods in geometry, Various Publ. Ser., vol. 30, Aarhus Univ., Aarhus, 1979, pp. 37–100. MR 552658
  • Michel Coste and Marie-Françoise Roy, La topologie du spectre réel, Ordered fields and real algebraic geometry (San Francisco, Calif., 1981), Contemp. Math., vol. 8, Amer. Math. Soc., Providence, R.I., 1982, pp. 27–59 (French). MR 653174
  • [CR] M.-F. Coste-Roy, Spectre réel d'un anneau et topos étale réel, Thèse Univ. Paris-Nord, 1980.
  • J.-L. Colliot-Thélène, Variantes du Nullstellensatz réel et anneaux formellement réels, Real algebraic geometry and quadratic forms (Rennes, 1981) Lecture Notes in Math., vol. 959, Springer, Berlin, 1982, pp. 98–108 (French). MR 683130, 10.1007/BFb0062251
  • Jean-Louis Colliot-Thélène, Michael Coste, Louis Mahé, and Marie-Françoise Roy (eds.), Géométrie algébrique réelle et formes quadratiques, Lecture Notes in Mathematics, vol. 959, Springer-Verlag, Berlin-New York, 1982 (French). MR 683126
  • Hans Delfs, The homotopy axiom in semialgebraic cohomology, J. Reine Angew. Math. 355 (1985), 108–128. MR 772485, 10.1515/crll.1985.355.108
  • Charles N. Delzell, A finiteness theorem for open semi-algebraic sets, with applications to Hilbert’s 17th problem, Ordered fields and real algebraic geometry (San Francisco, Calif., 1981), Contemp. Math., vol. 8, Amer. Math. Soc., Providence, R.I., 1982, pp. 79–97. MR 653176
  • Lou van den Dries, Some applications of a model theoretic fact to (semi-) algebraic geometry, Nederl. Akad. Wetensch. Indag. Math. 44 (1982), no. 4, 397–401. MR 683527
  • D. W. Dubois, Real commutative algebra I. Places, Rev. Mat. Hisp.-Amer. (4) 39 (1979), no. 2-3, 57–65. MR 629092
  • D. W. Dubois and G. Efroymson, Algebraic theory of real varieties. I, Studies and Essays (Presented to Yu-why Chen on his 60th Birthday, April 1, 1970) Math. Res. Center, Nat. Taiwan Univ., Taipei, 1970, pp. 107–135. MR 0280483
  • Hans Delfs and Manfred Knebusch, Semialgebraic topology over a real closed field. II. Basic theory of semialgebraic spaces, Math. Z. 178 (1981), no. 2, 175–213. MR 631628, 10.1007/BF01262039
  • Hans Delfs and Manfred Knebusch, On the homology of algebraic varieties over real closed fields, J. Reine Angew. Math. 335 (1982), 122–163. MR 667464, 10.1515/crll.1982.335.122
  • Hans Delfs and Manfred Knebusch, Locally semialgebraic spaces, Lecture Notes in Mathematics, vol. 1173, Springer-Verlag, Berlin, 1985. MR 819737
  • Donald Ward Dubois and Tómas Recio (eds.), Ordered fields and real algebraic geometry, Contemporary Mathematics, vol. 8, American Mathematical Society, Providence, R.I., 1982. MR 653171
  • Irving Kaplansky, Commutative rings, Revised edition, The University of Chicago Press, Chicago, Ill.-London, 1974. MR 0345945
  • Manfred Knebusch, An invitation to real spectra, Quadratic and Hermitian forms (Hamilton, Ont., 1983) CMS Conf. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 1984, pp. 51–105, 337–338. MR 776447
  • Tsit Yuen Lam, Orderings, valuations and quadratic forms, CBMS Regional Conference Series in Mathematics, vol. 52, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1983. MR 714331
  • T. Y. Lam, An introduction to real algebra, Rocky Mountain J. Math. 14 (1984), no. 4, 767–814. Ordered fields and real algebraic geometry (Boulder, Colo., 1983). MR 773114, 10.1216/RMJ-1984-14-4-767
  • Serge Lang, Algebraic structures, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. MR 0210527
  • [Lo] S. Lojasiewicz, Ensembles semi-analytiques, multigraphié I.H.E.S., 1965.
  • Kenneth McKenna, New facts about Hilbert’s seventeenth problem, Model theory and algebra (A memorial tribute to Abraham Robinson), Springer, Berlin, 1975, pp. 220–230. Lecture Notes in Math., Vol. 498. MR 0401720
  • Alexander Prestel, Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. MR 769847
  • Sibylla Prieß-Crampe, Angeordnete Strukturen, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 98, Springer-Verlag, Berlin, 1983 (German). Gruppen, Körper, projektive Ebenen. [Groups, fields, projective planes]. MR 704186
  • Alexander Prestel and Peter Roquette, Formally 𝑝-adic fields, Lecture Notes in Mathematics, vol. 1050, Springer-Verlag, Berlin, 1984. MR 738076
  • Marie-Françoise Coste-Roy, Faisceau structural sur le spectre réel et fonctions de Nash, Real algebraic geometry and quadratic forms (Rennes, 1981) Lecture Notes in Math., vol. 959, Springer, Berlin-New York, 1982, pp. 406–432 (French). MR 683146
  • Gerald E. Sacks, Saturated model theory, W. A. Benjamin, Inc., Reading, Mass., 1972. Mathematics Lecture Note Series. MR 0398817
  • Niels Schwartz, Der Raum der Zusammenhangskomponenten einer reellen Varietät, Geom. Dedicata 13 (1983), no. 4, 361–397 (German). MR 699882, 10.1007/BF00147538
  • Niels Schwartz, Real closed spaces, Rocky Mountain J. Math. 14 (1984), no. 4, 971–972. Ordered fields and real algebraic geometry (Boulder, Colo., 1983). MR 773144, 10.1216/RMJ-1984-14-4-971
  • N. Schwartz, Real closed rings, Algebra and order (Luminy-Marseille, 1984) Res. Exp. Math., vol. 14, Heldermann, Berlin, 1986, pp. 175–194. MR 891460
  • Gilbert Stengle, A nullstellensatz and a positivstellensatz in semialgebraic geometry, Math. Ann. 207 (1974), 87–97. MR 0332747
  • [W] B. L. v. d. Waerden, Algebra. I. 5th ed., Springer-Verlag, Berlin, Göttingen, Heidelberg, 1960.
  • Hassler Whitney, Elementary structure of real algebraic varieties, Ann. of Math. (2) 66 (1957), 545–556. MR 0095844
  • [ZS] O. Zariski and P. Samuel, Commutative algebra. II, New York, 1960.

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DOI: https://doi.org/10.1090/S0273-0979-1986-15431-5