Spectral spaces versus distributive lattices: a dictionary
Henri Lombardi
TL;DR
The paper develops a constructive dictionary between spectral spaces and distributive lattices, showing how Stone duality and Zariski-style spectra can be interpreted purely in lattice-theoretic terms via entailment relations. It extends the framework to encompass multiple spectra (real, valuative, linear, Heitmann/J-spectra) arising from dynamical algebraic structures, and introduces a constructive calculus (entailment theorems, Positivstellensatz variants) that yields computable content for classical results. It presents constructive versions of Kronecker, Forster-Swan, Serre, and Bass theorems, along with dimension theory and gluing results, and demonstrates the interaction between local data and global lattices through subspectral constructions and quotient gluing. The work thereby enables constructive proofs and applications in commutative algebra and algebraic geometry, with concrete notions of open maps and spectral-subspace correspondences, and highlights open problems in providing fully constructive proofs of classical theorems like Grothendieck’s open-map result.
Abstract
The category of distributive lattices is, in classical mathematics, antiequivalent to the category of spectral spaces. We give here some examples and a short dictionary for this antiequivalence. We propose a translation of several abstract theorems (in classical mathematics) into constructive ones, even in the case where points of a spectral space have no clear constructive content. La catégorie des treillis distributifs et celle des espaces spectraux sont antiéquivalentes (en mathématiques classiques). Nous proposons ici un petit dictionnaire pour cette antiéquivalence. Nous indiquons comment un certain nombre de théorèmes étranges des mathématiques classiques obtiennent un contenu constructif grâce à cette antiéquivalence, même dans le cas, fréquent, où les points des espaces spectraux considérés n'ont pas de contenu constructif clair.