On the Krull dimension of rings of semialgebraic functions

Abstract

Let $R$ be a real closed field and let $\mathcal{S}(M)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^n$ and let ${\mathcal{S}}^{\ast }(M)$ be its subring of bounded semialgebraic functions. In this work we introduce the concept of \em semialgebraic depth \em of a prime ideal $\gtp$ of $\mathcal{S}(M)$ in order to provide an elementary proof of the finiteness of the Krull dimension of the rings $\mathcal{S}(M)$ and ${\mathcal{S}}^{\ast }(M)$, inspired in the classical way of doing to compute the dimension of a ring of polynomials on a complex algebraic set and without involving the sophisticated machinery of real spectra. We also show that $\dim \mathcal{S}(M)=\dim {\mathcal{S}}^{\ast }(M)=\dim M$ and we 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 $\gtp$ is a prime \em $z$-ideal \em of $\mathcal{S}(M)$, its semialgebraic depth coincides with the transcendence degree over $R$ of the real closed field $\qf({\mathcal S}(M)/\gtp)$.

Cited by in corpus (4)

• On the substitution theorem for rings of semialgebraic functions
• On spectral types of semialgebraic sets
• On the remainder of the semialgebraic Stone-Cěch compactification of a semialgebraic set
• On the size of the fibers of spectral maps induced by semialgebraic embeddings