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arXiv · 2411.13706

Closed subcategories of quotient categories

Daniel Rogalski
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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.

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Daniel Rogalski. 2024-11-20. Closed subcategories of quotient categories. https://arxiv.org/abs/2411.13706

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