Finite intersection of valuation overrings of polynomial rings in at most three variables (Q6561512)

Let \(k\) be a field, \(D\) be any of the polynomial rings \(k[x_1]\), \(k[x_1,x_2]\), \(k[x_1,x_2,x_3]\), and \(K\) be the fraction field of \(D\). We say that a lattice-ordered group (\(\ell\)-group) \(G\) is realizable over \(D\) if there is a ring \(R\), \(D\subseteq R \subseteq K\) such that \(G\) is isomorphic to the group of divisibility \(G(R)\) of \(R\) (which is defined here as the multiplivative group of nonzero principal fractional ideals of \(R\)). The group \(G(R)\) is ordered by reverse inclusion. The aim of this paper is to classify all semi-local \(\ell\)-groups which are realized over \(D\) (semi-local means that it contains a finite number of maximal filters). The proofs rely on the study of the group of divisibility of finite intersections of valuation overrings of \(D\). We start with definitions. The cardinal product of two ordered groups \(G_1\) and \(G_2\), denoted by \(G_1\times_c G_2\), is the product of the groups \(G_1\) and \(G_2\) with pointwise ordering, and \(G_1\times_l G_2\) denotes this product together with the lexicographic order. \N\NIf \(D=k[x_1]\), then a nontrivial semilocal \(\ell\)-group \(G\) is realizable over \(D\) if, and only if, it is isomorphic (as \(\ell\)-group) to \(\mathbb{Z}\times_c \mathbb{Z}\times_c \cdots \times_c\mathbb{Z}\) for some finite number of copies of \(\mathbb{Z}\). \N\NIn the case where \(D=k[x_1,x_2]\), the author assumes that \(k\) is infinite. The condition is \(G\) being isomorphic to a cardinal product \(G_1\times_c G_2\times_c G_3\) of semilocal \(\ell\)-groups, where \(G_1\) is isomorphic to a finite cardinal product of subgroups of \(\mathbb{Q}\), \(G_2\) is isomorphic to a finite cardinal product of finitely generated subgroups of \(\mathbb{R}\) having rational rank \(2\), and \(G_3\) is isomorphic to a finite cardinal product of \(\ell\)-groups of the form \(\mathbb{Z}\times_l(\mathbb{Z}\times_c\cdots\times_c\mathbb{Z})\). \N\NTurning to \(D=k[x_1,x_2,x_3]\), the author still assumes that \(k\) is infinite and he characterizes groups which are weakly realizable. This case requires more definitions. If \(0\rightarrow A \stackrel{\alpha}{\rightarrow} B\stackrel{\beta}{\rightarrow} C \rightarrow 0\) is an exact sequence such that \(\{ b\in B\; :\; b\geq 0\}=\{b\in B\; :\; \beta(b)>0\}\cup \{\alpha(a)\; :\; a\in A,\; a\geq 0\}\), then \(B\) is called a lex-extension of \(A\) by \(C\). A lexico-cardinal decomposition of \(G\) is a decomposition of the form \[\displaystyle{G=\sideset{}{_c}\prod_{\sigma_1\in\Lambda(\sigma_0)}\left(\mbox{lex extension of }\left[\;\sideset{}{_c}\prod_{\sigma_2\in\Lambda(\sigma_1)} \left(\mbox{lex extension of }\left[\;\sideset{}{_c}\prod_{\sigma_3\in\Lambda(\sigma_2)}\right(\; \cdots \right. \right. \right.}\] \[\displaystyle{\left.\left.\left.\left.\left.\left(\mbox{lex extension of }\left[\;\sideset{}{_c}\prod_{\sigma_{d+1}\in\Lambda(\sigma_d)} H_{\sigma_d,\sigma_{d+1}}\right]\mbox{by }H_{\sigma_{d-1},\sigma_d}\right)\cdots\right)\right]\mbox{by }H_{\sigma_1,\sigma_2}\right)\right]\mbox{by } H_{\sigma_0,\sigma_1}\right), }\] where \(\sideset{}{_c}\prod\) denotes the cardinal product, \(d\in \mathbb{N}\), \(\sigma_0\) is an index, \(\Lambda(\sigma_0)\) is a non-empty finite set, for \(i\in \{1,\dots ,d\}\) \(\Lambda(\sigma_i)\) is a finite set wich is either empty or of cardinal greater than \(1\), and \(H_{\sigma_{i-1},\sigma_i}\) is a non-trivial totally ordered group. We say that \(G\) is weakly realizable over \(D\) if there exists a Bézout overring \(R\) of \(D\) (\(R\subseteq K\)) such that \(G\) and \(G(R)\) admit a lexico-cardinal decomposition of the same form. \N\NThe author proves that \(G\) is weakly realizable over \(D\) if, and only if, \(G\) is isomorphic to a group of the form \(G_1\times_c G_2\times_c G_3\times_c G_4\times_c G_5\times_c G_6\), where: \N\begin{itemize}\N\item \(G_1\) is isomorphic to \(\mathbb{Z}\times_l A\), where \(A\) is a finite cardinal product of one or more of the following groups: \N\begin{itemize}\N\item[-] a finite cardinal product of \(\mathbb{Z}+r_i\mathbb{Z}\)'s (\(r_i\) irrational numbers), \\\N\item[-] a finite cardinal product of subgroups of \(\mathbb{Q}\), \\\N\item[-] some \(\mathbb{Z}\times_l(\mathbb{Z}\times_c\cdots\times_c\mathbb{Z})\), \N\end{itemize}\N\item \(G_2\) is a finite cardinal product of lex-extensions of a finite cardinal product of copies of \(\mathbb{Z}\) by a not finitely generated subgroup of \(\mathbb{Q}\), \N\item \(G_3\) is isomorphic to a finite cardinal product of groups of the form \((\mathbb{Z}+r\mathbb{Z})\times_l(\mathbb{Z}\times_c\cdots\times_c\mathbb{Z})\), \N\item each of \(G_4\), \(G_5\) and \(G_6\) is isomorphic to a cardinal sum of subgroups of \(\mathbb{R}\) of rational rank \(1\), \(2\) and \(3\) respectively.\N\end{itemize}