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Valuative Lattices and Spectra

In: Algebraic, Number Theoretic, and Topological Aspects of Ring Theory

Author

Listed:
  • Henri Lombardi

    (Université de Franche-Comté)

  • Assia Mahboubi

    (École Centrale Nantes, CNRS, INRIA, LS2N, UMR 6004, Nantes Université)

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 uncover 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 valued discrete 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 Valuativstellensätze corresponding to these theories, and we compare the various resulting notions of valuative dimensions.

Suggested Citation

  • Henri Lombardi & Assia Mahboubi, 2023. "Valuative Lattices and Spectra," Springer Books, in: Jean-Luc Chabert & Marco Fontana & Sophie Frisch & Sarah Glaz & Keith Johnson (ed.), Algebraic, Number Theoretic, and Topological Aspects of Ring Theory, pages 275-341, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-28847-0_17
    DOI: 10.1007/978-3-031-28847-0_17
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