Revista Matemática Iberoamericana

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Volume 31, Issue 3, 2015, pp. 753–766
DOI: 10.4171/RMI/852

Published online: 2015-10-29

On the Krull dimension of rings of continuous semialgebraic functions

José F. Fernando[1] and José Manuel Gamboa[2]

(1) Universidad Complutense de Madrid, Spain
(2) Universidad Complutense de Madrid, Spain

Let $R$ be a real closed field, ${\mathcal S}(M)$ the ring of continuous semialgebraic functions on a semialgebraic set $M\subset R^m$ and ${\mathcal S}^*(M)$ 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 semi algebraic depth of a prime ideal $\mathfrak p$ of ${\mathcal S}(M)$ in order to provide an elementary proof of the finiteness of the Krull dimensions of the rings ${\mathcal S}(M)$ and ${\mathcal S}^*(M)$ 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 $\mathrm {dim}({\mathcal S}(M))=\mathrm {dim}({\mathcal S}^*(M))=\mathrm {dim}(M)$ and prove that in both cases the height of a maximal ideal corresponding to a point $p \in M$ coincides with the local dimension of $M$ at $p$. In case $\mathfrak p$ is a prime $z$-ideal of ${\mathcal S}(M)$, its semialgebraic depth coincides with the transcendence degree of the real closed field $\mathrm {qf}({\mathcal S}(M)/\mathfrak p)$ over $R$.

Keywords: Continuous semialgebraic function, bounded continuous semialgebraic function, $z$-ideal, semialgebraic depth, Krull dimension, local dimension, transcendence degree, real closed field

Fernando José, Gamboa José Manuel: On the Krull dimension of rings of continuous semialgebraic functions. Rev. Mat. Iberoam. 31 (2015), 753-766. doi: 10.4171/RMI/852