Realizations of semilocal ℓ-groups over k[x₁,x₂,…x_n]

Abstract

Doering and Lequain in 1999 introduced a weak approximation theorem for dependent valuation rings and they proved that every finitely generated lattice-ordered group can be realized as the group of divisibility of a semilocal Bézout overring of a polynomial ring over a field k in infinitely many variables, where each of the valuation rings appearing in the finite intersection has residue field k. Moreover, they proved that every semilocal lattice-ordered group admits a lexico-cardinal decomposition form. In this work, we focus on realizing the semilocal $\mathcal{l}$-group over a polynomial ring in finitely many variables. We prove that every semilocal lattice-ordered group having a finite rational rank can be realized as the group of divisibility of a Bézout overring of $k[x_1,x_2,\dots ,x_n]$ up to lexico-cardinal decomposition, where k is a field and $x_1,x_2,\dots ,x_n$ are indeterminates over k and n depends on the group. As a corollary, we prove that every semilocal $\mathcal{l}$-group either finitely generated or divisible with finite rational rank is realizable over $k[x_1,x_2,\dots ,x_n],$ where each of the valuation rings appearing in the finite intersection has residue field k.

Keywords:

• Finite intersection
• group of divisibility
• lattice-ordered group
• overring
• valuation domain

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13F05
• 13F30
• Secondary: 13B22
• 14A15

Acknowledgements

I would like to express the deepest gratitude to Bruce Olberding for suggesting this topic. I thank him for the helpful discussions and comments that made to achieve the aim of this article.