The Boolean spectrum of a Grothendieck category
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.
Cite
@article{arxiv.2411.16309,
title = {The Boolean spectrum of a Grothendieck category},
author = {Henning Krause},
journal= {arXiv preprint arXiv:2411.16309},
year = {2024}
}
Comments
26 pages. V2: Substantial revision, including a new section on coproduct decompositions