On the size of the fibers of spectral maps induced by semialgebraic embeddings

Abstract

Let $\mathcal{S}(M)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset {\mathbb{R}}^m$ and ${\mathcal{S}}^{\ast }(M)$ its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps $\mathrm{Spec}(\mathtt{j}{)}_1:\mathrm{Spec}(\mathcal{S}(N))\to \mathrm{Spec}(\mathcal{S}(M))$ and $\mathrm{Spec}(\mathtt{j}{)}_2:\mathrm{Spec}({\mathcal{S}}^{\ast }(N))\to \mathrm{Spec}({\mathcal{S}}^{\ast }(M))$ induced by the inclusion $\mathtt{j}:N\hookrightarrow M$ of a semialgebraic subset $N$ of $M$. The ring $\mathcal{S}(M)$ can be understood as the localization of ${\mathcal{S}}^{\ast }(M)$ at the multiplicative subset ${\mathcal{W}}_M$ of those bounded semialgebraic functions on $M$ with empty zero set. This provides a natural inclusion ${\mathfrak{i}}_M:\mathrm{Spec}(\mathcal{S}(M))\hookrightarrow \mathrm{Spec}({\mathcal{S}}^{\ast }(M))$ that reduces both problems above to an analysis of the fibers of the spectral map $\mathrm{Spec}(\mathtt{j}{)}_2:\mathrm{Spec}({\mathcal{S}}^{\ast }(N))\to \mathrm{Spec}({\mathcal{S}}^{\ast }(M))$. If we denote $Z:={\mathrm{cl}}_{\mathrm{Spec}({\mathcal{S}}^{\ast }(M))}(M\setminus N)$, it holds that the restriction map $\mathrm{Spec}(\mathtt{j}{)}_2|:\mathrm{Spec}({\mathcal{S}}^{\ast }(N))\setminus \mathrm{Spec}(\mathtt{j}{)}_2^{-1}(Z)\to \mathrm{Spec}({\mathcal{S}}^{\ast }(M))\setminus Z$ is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of $\mathrm{Spec}(\mathtt{j}{)}_2$ at the points of $Z$. The size of the fibers of prime ideals `close' to the complement $Y:=M\setminus N$ provides valuable information concerning how $N$ is immersed inside $M$. If $N$ is dense in $M$, the map $\mathrm{Spec}(\mathtt{j}{)}_2$ is surjective and the generic fiber of a prime ideal $\mathfrak{p}\in Z$ contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber $\mathrm{Spec}(\mathtt{j}{)}_2^{-1}(\mathfrak{p})$ is a finite set for $\mathfrak{p}\in Z$.

33 pages, 3 figures

References in corpus (1)

• On the Krull dimension of rings of semialgebraic functions