Integrality of noetherian Grothendieck categories
arXiv:1711.06946 · doi:10.1016/j.jalgebra.2021.10.036
Abstract
We introduce the notion of integrality of Grothendieck categories as a simultaneous generalization of the primeness of noncommutative noetherian rings and the integrality of locally noetherian schemes. Two different spaces associated to a Grothendieck category yield respective definitions of integrality, and we prove the equivalence of these definitions using a Grothendieck-categorical version of Gabriel's correspondence, which originally related indecomposable injective modules and prime two-sided ideals for noetherian rings. The generalization of prime two-sided ideals is also used to classify locally closed localizing subcategories. As an application of the main results, we develop a theory of singular objects in a Grothendieck category and deduce Goldie's theorem on the existence of the quotient ring as its consequence.
46 pages