Closed subcategories of quotient categories

TL;DR

This paper explores the structure of closed subcategories in quotient categories of Grothendieck categories, linking them to geometric concepts like subschemes, and provides a framework for understanding their classification.

Contribution

It introduces a method to describe closed subcategories of quotient categories using properties of closed subcategories in the original category, especially when it has compact projective generators.

Findings

Closed subcategories correspond to closed subschemes in the commutative case.

Provides a description of closed subcategories in quotient categories.

Applicable to noncommutative projective schemes like Qgr-B.

Abstract

We study the spectrum of closed subcategories in a quasi-scheme, i.e. a Grothendieck category $X$. The closed subcategories are the direct analogs of closed subschemes in the commutative case, in the sense that when $X$ is the category of quasi-coherent sheaves on a quasi-projective scheme $S$, then the closed subschemes of $S$ correspond bijectively to the closed subcategories of $X$. Many interesting quasi-schemes, such as the noncommutative projective scheme Qgr-$B$ = Gr-$B$/Tors-$B$ associated to a graded algebra $B$, arise as quotient categories of simpler abelian categories. In this paper we will show how to describe the closed subcategories of any quotient category $X/Y$ in terms of closed subcategories of $X$ with special properties, when $X$ is a category with a set of compact projective generators.