Specialization orders on atom spectra of Grothendieck categories

TL;DR

This paper develops methods to construct Grothendieck categories from colored quivers and demonstrates that any partial order can be realized as the atom spectrum of such a category, extending Hochster's classical result.

Contribution

It introduces a systematic construction of Grothendieck categories from colored quivers and characterizes their atom spectra, including realization of arbitrary partial orders.

Findings

Any partial order can be realized as the atom spectrum of a Grothendieck category.

Existence of Grothendieck categories with empty atom spectrum but nonempty injective spectrum.

Abstract

We introduce systematic methods to construct Grothendieck categories from colored quivers and develop a theory of the specialization orders on the atom spectra of Grothendieck categories. We show that any partially ordered set is realized as the atom spectrum of some Grothendieck category, which is an analog of Hochster's result in commutative ring theory. We also show that there exists a Grothendieck category which has empty atom spectrum but has nonempty injective spectrum.