Specialization orders on atom spectra of Grothendieck categories

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.

39 pages

References in corpus (1)

• Extension groups between atoms and objects in locally noetherian Grothendieck category

Cited by in corpus (8)

• Classification of categorical subspaces of locally noetherian schemes
• Finiteness of the number of minimal atoms in Grothendieck categories
• Integrality of noetherian Grothendieck categories
• Construction of Grothendieck categories with enough compressible objects using colored quivers
• Classifying subcategories, local algebra and the Rosenberg spectrum of a locally noetherian category
• Extension groups between atoms in abelian categories
• On cosilting hearts over the Kronecker algebra
• A new dimension for Grothendieck categories via the atom spectrum