Heitmann dimension of distributive lattices and commutative rings
Thierry Coquand, Henri Lombardi, Claude Quitté
TL;DR
The paper presents a constructive treatment of Heitmann dimension for distributive lattices and commutative rings, framing the theory through spectral spaces and the Zariski lattice. It defines two complementary dimensions, the Krull- and Heitmann-dimensions, via boundary operations on lattices and through the Heitmann lattice $\mathrm{He}(\mathbf{T})$, establishing that $\mathsf{Hdim}\mathbf{T}\le\mathsf{Jdim}\mathbf{T}$ and relating $\mathsf{Jdim}$ to the Jacobson-closure of maximal spectra. The work builds a constructive bridge from lattice theory to commutative algebra by introducing the Zariski lattice $\mathsf{Zar}\mathbf{A}$ and its Heitmann counterpart, enabling dimension-theoretic results without reliance on choice or LEM. It also outlines constructive proofs of key non-Noetherian theorems (Swan, Serre) in this framework and clarifies how spectral-subspace glueing and quotients interact with Krull/Heitmann dimensions. Overall, the paper provides a modular, point-free, algorithm-friendly approach to dimension theory in distributive lattices and rings, with concrete definitions and comparatives between $\mathsf{Kdim}$, $\mathsf{Jdim}$, and $\mathsf{Hdim}$ that illuminate constructive aspects of classical results.
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 Mathématiques de Besançon. Algèbre et théorie 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 Quitté. 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 corrigée des 4 premières sections de l'article ``Dimension de Heitmann des treillis distributifs et des anneaux commutatifs. Publications Mathématiques de Besançon. Algèbre et théorie des nombres, 2006'' Les sections 5 à 7 de l'article original sont traitées de manière un peu plus simple dans ``Henri Lombardi and Claude Quitté. Commutative algebra: constructive methods. Finite projective modules. Springer, 2015.'' Nous étudions 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 démonstrations. Nous développons ceci d'abord dans le cadre général de la théorie des treillis distributifs et des espaces spectraux. Nous appliquons ensuite cette problématique dans le cadre de l'algèbre commutative.