Publication: On the Krull dimension of rings of continuous semialgebraic functions
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2015
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Universidad Autónoma Madrid
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Let R be a real closed field, S(M) the ring of continuous semialgebraic functions on a semialgebraic set M subset of R-m and S* (M) its subring of continuous semialgebraic functions that are bounded with respect to R. In this work we introduce semialgebraic pseudo-compactifications of M and the semialgebraic depth of a prime ideal p of S(M) in order to provide an elementary proof of the finiteness of the Krull dimensions of the rings S(M) and S* (M) for an arbitrary semialgebraic set M. We are inspired by the classical way to compute the dimension of the ring of polynomial functions on a complex algebraic set without involving the sophisticated machinery of real spectra. We show dim(S(M)) = dim(S* (M)) = dim(M) and prove that in both cases the height of a maximal ideal corresponding to a point p is an element of M coincides with the local dimension of M at p. In case p is a prime z-ideal of S(M), its semialgebraic depth coincides with the transcendence degree of the real closed field qf(S(M)/p) over R
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J. Atiyah, I.G. MacDonald: Introduction to Commutative Algebra. Addison–Wesley Publishing Co., Inc., Reading, Mass.-London: 1969.
J. Bochnak, M. Coste, M.-F. Roy: Real algebraic geometry. Ergeb. Math. 36, Springer-Verlag, Berlin: 1998.
M. Carral, M. Coste: Normal spectral spaces and their dimensions. J. Pure Appl. Algebra 30 (1983), 227-235.
G.L. Cherlin, M.A. Dickmann: Real closed rings. I. Residue rings of rings of continuous functions. Fund. Math. 126 (1986), no. 2, 147–183.
G.L. Cherlin, M.A. Dickmann: Real closed rings. II. Model theory. Ann. Pure Appl. Logic 25 (1983), no. 3, 213–231.
H. Delfs, M. Knebusch: Separation, Retractions and homotopy extension in semialgebraic spaces. Pacific J. Math. 114 (1984), no. 1, 47–71.
H. Delfs, M. Knebusch: Locally semialgebraic spaces. Lecture Notes in Mathematics, 1173. SpringerVerlag, Berlin: 1985.
J.M. Gamboa: On prime ideals in rings of semialgebraic functions. Proc. Amer. Math. Soc. 118 (1993), no. 4, 1034–1041.
J.M. Gamboa, J.M. Ruiz: On rings of semialgebraic functions. Math. Z. 206 (1991) no. 4, 527–532.
L. Gillman, M. Jerison: Rings of continuous functions. The Univ. Series in Higher Nathematics 1, D. Van Nostrand Company, Inc.: 1960.
A. Prestel, N. Schwartz: Model theory of real closed rings. Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 261–290, Fields Inst. Commun., 32, Amer. Math. Soc., Providence, RI, 2002.
N. Schwartz: Real closed rings. Habilitationsschrift, München: 1984.
N. Schwartz: Real closed rings. Algebra and order (Luminy-Marseille, 1984), 175–194, Res. Exp. Math., 14, Heldermann, Berlin: 1986.
N. Schwartz: The basic theory of real closed spaces. Mem. Amer. Math. Soc. 77 (1989), no. 397, viii + 122 pp.
N. Schwartz: Rings of continuous functions as real closed rings. Ordered algebraic structures (Cura¸cao, 1995), 277–313, Kluwer Acad. Publ., Dordrecht: 1997.
N. Schwartz: Epimorphic extensions and Prüfer extensions of partially ordered rings. Manuscripta Math. 102 (2000), 347-381.
N. Schwartz, J.J. Madden: Semi-algebraic function rings and reflectors of partially ordered rings. Lecture Notes in Mathematics, 1712. Springer-Verlag, Berlin: 1999. xii+279 pp.
N. Schwartz, M. Tressl: Elementary properties of minimal and maximal points in Zariski spectra. J. Algebra 323 (2010), no. 3, 698–728.
M. Tressl: The real spectrum of continuous definable functions in o-minimal structures. S´eminaire de Structures Algébriques Ordonnées 1997-1998, 68, Mars 1999, p. 1-15.
M. Tressl: Super real closed rings. Fund. Math. 194 (2007), no. 2, 121–177.
M. Tressl: Bounded super real closed rings. Logic Colloquium 2007, 220–237, Lect. Notes Log., 35, Assoc. Symbol. Logic, La Jolla, CA, 2010.