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oai:docta.ucm.es:20.500.14352/349732024-03-01T13:16:09Zcom_20.500.14352_14col_20.500.14352_15

On the Krull dimension of rings of continuous semialgebraic functions Fernando Galván, José Francisco Gamboa Mutuberria, José Manuel 512.7 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 Geometria algebraica 1201.01 Geometría Algebraica 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 GAAR Depto. de Álgebra, Geometría y Topología Fac. de Ciencias Matemáticas TRUE pub 2023-06-19T14:57:54Z 2023-06-19T14:57:54Z 2015 journal article https://hdl.handle.net/20.500.14352/34973 0213-2230 10.4171/RMI/852 eng MTM2011-22435 open access application/pdf Universidad Autónoma Madrid