An analogue of Stone duality via support

TL;DR

This paper explores a lattice-theoretic analogue of Stone duality through support, connecting lattices and topological spaces, with implications for triangulated categories and tensor exact categories.

Contribution

It introduces a support-based framework that parallels Stone duality, clarifying the relationship between lattices and topological spaces in a new lattice-theoretic context.

Findings

Support provides an analogue of Stone duality relating lattices and topological spaces.

The parallel between support via closed and open sets is explained using Hochster duality.

Implications for tensor exact categories are discussed.

Abstract

The notion of support provides an analogue of Stone duality, relating lattices to topological spaces. This note aims to explain in lattice theoretic terms what has been developed in the context of triangulated categories. In particular, the parallel between support via closed and open sets is addressed in terms of Hochster duality. As an application we indicate some consequences for tensor exact categories.