Valuative lattices and spectra

TL;DR

This paper surveys dynamical methods in mathematics that reveal computational content in abstract objects and applies these methods to compare and unify different notions of valuative spectra in algebraic geometry.

Contribution

It introduces a new notion of valuative spectrum, compares existing notions, and establishes formal Valuativestellens"atze within the dynamical framework.

Findings

Valuative lattices from different theories are essentially the same.

Dynamical methods uncover algorithms from classical proofs.

Comparison of valuative dimensions in different theories.

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.