On the Krull dimension of rings of continuous semialgebraic functions

Abstract

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 ${\mathcal{S}}^{\ast }(M)$ 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 ${\mathcal{S}}^{\ast }(M)$ 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 $\dim (\mathcal{S}(M))=\dim ({\mathcal{S}}^{\ast }(M))=\dim (M)$ 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 $R$.