Constructions cach\'ees en alg\`ebre abstraite. Dimension de Krull, Going Up, Going Down (revised version, 2008)

TL;DR

This paper develops constructive versions of Krull's dimension theory for rings and lattices, enabling explicit computation and a partial realization of Hilbert's program in classical algebra.

Contribution

It introduces constructive approaches to classical theorems in Krull dimension theory, allowing explicit computation and applications to concrete objects.

Findings

Constructive versions of Krull's dimension theory for rings and lattices.

Explicit computational content derived from classical theorems.

Application to a partial realization of Hilbert's program in algebra.

Abstract

Nous rappelons des versions constructives de la th\'eorie de la dimension de Krull dans les anneaux commutatifs et dans les treillis distributifs, dont les bases ont \'et\'e pos\'ees par Joyal, Espan\~ol et les deux auteurs. Nous montrons sur les exemples de la dimension des alg\`ebres de pr\'esentation finie, du Going Up, du Going Down \ldots que cela nous permet de donner une version constructive de grands th\'eor\`emes classiques, et par cons\'equent de r\'ecup\'erer un contenu calculatoire explicite lorsque ces th\'eor\`emes abstraits sont utilis\'es pour d\'emontrer l'existence d'objets concrets. Nous pensons ainsi mettre en oeuvre une r\'ealisation partielle du programme de Hilbert pour l'alg\`ebre abstraite classique. We present constructive versions of Krull's dimension theory for commutative rings and distributive lattices. The foundations of these constructive versions are due to Joyal, Espan\~ol and the authors. We show that this gives a constructive version of basic classical theorems (dimension of finitely presented algebras, Going up and Going down theorem, \ldots), and hence that we get an explicit computational content when these abstract results are used to show the existence of concrete elements. This can be seen as a partial realisation of Hilbert's program for classical abstract commutative algebra.