Valuative lattices and spectra
Abstract
The first part of the present article consists in a survey about the dynamical constructive method designed using dynamical theories and dynamical algebraic structures. Dynamical methods uncovers a hidden computational content for numerous abstract objects of classical mathematics, which seem a priori inaccessible constructively, e.g., the algebraic closure of a (discrete) field. When a proof in classical mathematics uses these abstract objects and results in a concrete outcome, dynamical methods generally make possible to discover an algorithm for this concrete outcome. The second part of the article applies this dynamical method to the theory of divisibility. We compare two notions of valuative spectra present in the literature and we introduce a third notion, which is implicit in an article devoted to the dynamical theory of algebraically closed discrete valued fields. The two first notions are respectively due to Huber \& Knebusch and to Coquand. We prove that the corresponding valuative lattices are essentially the same. We establish formal Valuativestellens\"atze corresponding to these theories, and we compare the various resulting notions of valuative dimensions.
Cite
@article{arxiv.2210.16558,
title = {Valuative lattices and spectra},
author = {Henri Lombardi and Assia Mahboubi},
journal= {arXiv preprint arXiv:2210.16558},
year = {2023}
}
Comments
This file contains also a French version of the paper. English version appears in the Proceedings of Graz Conference on Rings and Factorizations 2021. Title: Algebraic, Number Theoretic, and Topological Aspects of Ring Theory. Editors: Jean-Luc Chabert, Marco Fontana, Sophie Frisch, Sarah Glaz, Keith Johnson. Springer 2023 ISBN 978-3-031-28846-3 DOI 10.1007/978-3-031-28847-0