The Boolean spectrum of a Grothendieck category

TL;DR

This paper introduces a support theory for objects in Grothendieck categories based on Boolean lattices, enabling classification of subcategories and decompositions, extending known correspondences in module theory.

Contribution

It develops a new support notion using spectral categories and Boolean lattices, providing a framework for classifying subcategories and decompositions in Grothendieck categories.

Findings

Classifies subcategories closed under coproducts, subobjects, and extensions.

Describes coproduct decompositions via Boolean lattices.

Extends Crawley-Boevey's correspondence to Grothendieck categories.

Abstract

A notion of support for objects in any Grothendieck category is introduced. This is based on the spectral category of a Grothendieck category and uses its Boolean lattice of localising subcategories. The support provides a classification of all subcategories that are closed under arbitrary coproducts, subobjects, and essential extensions. There is also a notion of exact support which classifies certain thick subcategories. As an application, the coproduct decompositions of objects are described in terms of Boolean lattices. Also, for any ring Crawley-Boevey's correspondence between definable subcategories of modules and closed subsets of the Ziegler spectrum is extended.