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Specialization orders on atom spectra of Grothendieck categories. (English) Zbl 1417.18003

Summary: 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.

MSC:

18E15 Grothendieck categories (MSC2010)
16G30 Representations of orders, lattices, algebras over commutative rings
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