JournalsrmiVol. 31, No. 3pp. 753–766

On the Krull dimension of rings of continuous semialgebraic functions

  • José F. Fernando

    Universidad Complutense de Madrid, Spain
  • José Manuel Gamboa

    Universidad Complutense de Madrid, Spain
On the Krull dimension of rings of continuous semialgebraic functions cover
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Abstract

Let RR be a real closed field, S(M){\mathcal S}(M) the ring of continuous semialgebraic functions on a semialgebraic set MRmM\subset R^m and S(M){\mathcal S}^*(M) its subring of continuous semialgebraic functions that are bounded with respect to RR. In this work we introduce semialgebraic pseudo-compactifications of MM and the semi algebraic depth of a prime ideal p\mathfrak p of S(M){\mathcal S}(M) in order to provide an elementary proof of the finiteness of the Krull dimensions of the rings S(M){\mathcal S}(M) and S(M){\mathcal S}^*(M) for an arbitrary semialgebraic set MM. 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)\mathrm {dim}({\mathcal S}(M))=\mathrm {dim}({\mathcal S}^*(M))=\mathrm {dim}(M) and prove that in both cases the height of a maximal ideal corresponding to a point pMp \in M coincides with the local dimension of MM at pp. In case p\mathfrak p is a prime zz-ideal of S(M){\mathcal S}(M), its semialgebraic depth coincides with the transcendence degree of the real closed field qf(S(M)/p)\mathrm {qf}({\mathcal S}(M)/\mathfrak p) over RR.

Cite this article

José F. Fernando, José Manuel Gamboa, On the Krull dimension of rings of continuous semialgebraic functions. Rev. Mat. Iberoam. 31 (2015), no. 3, pp. 753–766

DOI 10.4171/RMI/852