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arXiv · 2312.00684

Heitmann dimension of distributive lattices and commutative rings

Thierry Coquand·Henri Lombardi·Claude Quitté
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Abstract

This paper is the English translation of the first 4 sections of the article ``Dimension de Heitmann des treillis distributifs et des anneaux commutatifs. Publications Math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres, 2006'', after some corrections. Sections 5-7 of the original article are treated a bit more simply in the book ``Henri Lombardi and Claude Quitt\'e. Commutative algebra: constructive methods. Finite projective modules. Springer, 2015.'' We study the notion of dimension introduced by Heitmann in his remarkable article ``Generating non-Noetherian modules efficiently, Mich. Math. J., 31, (1084)'' as well as a related notion, only implicit in his proofs. We first develop this within the general framework of the theory of distributive lattices and spectral spaces. -- Cet article est une version corrig\'ee des 4 premi\`eres sections de l'article ``Dimension de Heitmann des treillis distributifs et des anneaux commutatifs. Publications Math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres, 2006'' Les sections 5 \`a 7 de l'article original sont trait\'ees de mani\`ere un peu plus simple dans ``Henri Lombardi and Claude Quitt\'e. Commutative algebra: constructive methods. Finite projective modules. Springer, 2015.'' Nous \'etudions la notion de dimension introduite par Heitmann dans son article remarquable ``Generating non-Noetherian modules efficiently, Mich. Math. J., 31, (1084)'', ainsi qu'une notion voisine, seulement implicite dans ses d\'emonstrations. Nous d\'eveloppons ceci d'abord dans le cadre g\'en\'eral de la th\'eorie des treillis distributifs et des espaces spectraux. Nous appliquons ensuite cette probl\'ematique dans le cadre de l'alg\`ebre commutative.

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Thierry Coquand, Henri Lombardi, Claude Quitté. 2023-12-01. Heitmann dimension of distributive lattices and commutative rings. https://arxiv.org/abs/2312.00684

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