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On rings of semialgebraic functions

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1991
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Springer Verlag
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The authors study some properties of the ring of abstract semialgebraic functions over a constructible subset of the real spectrum of an excellent ring. To be more precise, let X be a constructible subset of the real spectrum of a ring A. The ring S(X) of abstract semialgebraic functions over X was introduced bz N. Schwartz [see Mem. Am. Math. Soc. 397 (1989; Zbl 0697.14015)], as a generalization of continuous functions with semialgebraic graph to the context of real spectra. Unfortunately the utility of this functions is not yet quite established. The main result of the paper states that if A is excellent, the Krull dimension of S(X) equals the dimension of X (defined as the maximum of the heights of the supports of points lying in X), which in turn, as J. M. Ruiz showed in C. R. Acad. Sci. Paris, S´er. I 302, 67-69 (1986; Zbl 0591.13017) coincides with its topological dimension. This was first shown by M. Carral and M. Coste [J. Pure Appl. Algebra 30, 227-235 (1983; Zbl 0525.14015)] for the particular case of X being a ‘true’ semialgebraic subset which is locally closed, and the result extends readily to abstract locally closed constructible sets. Then the authors use the compactness of the constructible topology of real spectra and the properties of excellent rings to reduce the general case to the locally closed one. The paper finishes by characterizing the finitely generated prime ideals of S(X), namely they are the ideals of the open constructible points of X whose closure in X is open of dimension 6= 1.
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