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arXiv · 1711.06946

Integrality of noetherian Grothendieck categories

Ryo Kanda
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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.

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Ryo Kanda. 2022-03-22. Integrality of noetherian Grothendieck categories. https://doi.org/10.1016/j.jalgebra.2021.10.036

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