Extension groups between atoms in abelian categories

TL;DR

This paper introduces extension groups between atoms in abelian categories, characterizes certain subcategories, and explores their topological and homological properties, revealing new insights into the structure of atom spectra.

Contribution

It defines extension groups between atoms, introduces virtual duals to measure global dimension, and uncovers a new topological property of atom spectra.

Findings

Characterization of localizing subcategories via extension groups

Introduction of virtual duals to assess global dimension

Existence of spectral spaces not homeomorphic to any atom spectrum

Abstract

We introduce the extension groups between atoms in an abelian category. For a locally noetherian Grothendieck category, the localizing subcategories closed under injective envelopes are characterized in terms of those extension groups. We also introduce the virtual duals of the extension groups between atoms to measure the global dimension of the category. A new topological property of atom spectra is revealed and it is used to relate the projective dimensions of atoms with the Krull-Gabriel dimensions. As a byproduct of the topological observation, we show that there exists a spectral space that is not homeomorphic to the atom spectrum of any abelian category.