The Boolean spectrum of a Grothendieck category

Henning Krause

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arXiv · 2411.16309

The Boolean spectrum of a Grothendieck category

Henning Krause

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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.

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Henning Krause. 2024-11-25. The Boolean spectrum of a Grothendieck category. https://arxiv.org/abs/2411.16309

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