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On the Krull dimension of rings of continuous semialgebraic functions

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Fernando Galván, José Francisco and Gamboa, J. M. (2015) On the Krull dimension of rings of continuous semialgebraic functions. Revista Matemática Iberoamericana, 31 (3). pp. 753-766. ISSN 0213-2230

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Official URL: http://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=31&iss=3&rank=1




Abstract

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


Item Type:Article
Uncontrolled Keywords:Semialgebraic function, bounded semialgebraic function, z-ideal, semialgebraic depth, Krull dimension, local dimension, transcendence degree, real closed ring, real closed field, real closure of a ring
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:34811
Deposited On:18 Dec 2015 09:18
Last Modified:02 Mar 2016 14:33

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