Distributive lattices and cohomology

TL;DR

This paper constructs a resolution for subgroup intersections in abelian groups with distributive lattices, applying it to module singularities, and deriving a generalized Chinese remainder theorem linked to Gelfand-Naimark duality.

Contribution

It introduces a novel resolution method for subgroup intersections in distributive lattice settings and connects it to classical theorems and dualities.

Findings

Resolution of subgroup intersections in distributive lattices

Application to detecting module singularities over Dedekind rings

Generalized Chinese remainder theorem derived

Abstract

A resolution of the intersection of a finite number of subgroups of an abelian group by means of their sums is constructed, provided the lattice generated by these subgroups is distributive. This is used for detecting singularities of modules over Dedekind rings. A generalized Chinese remainder theorem is derived as a consequence of the above resolution. The Gelfand-Naimark duality between finite closed coverings of compact Hausdorff spaces and the generalized Chinese remainder theorem is clarified.