Extension groups between atoms and objects in locally noetherian Grothendieck category

TL;DR

This paper introduces a new way to measure extension groups between atoms and objects in locally noetherian Grothendieck categories, linking them to Bass numbers and classifying certain subcategories.

Contribution

It defines the extension group as a module over a skew field and establishes a correspondence between subcategories and subsets of the atom spectrum.

Findings

Extension groups coincide with Bass numbers.

Bijection between E-stable subcategories and atom spectrum subsets.

Subcategories are closed under extensions, kernels, and cokernels.

Abstract

We define the extension group between an atom and an object in a locally noetherian Grothendieck category as a module over a skew field. We show that the dimension of the i-th extension group between an atom and an object coincides with the i-th Bass number of the object with respect to the atom. As an application, we give a bijection between the E-stable subcategories closed under arbitrary direct sums and direct summands and the subsets of the atom spectrum and show that such subcategories are also closed under extensions, kernels of epimorphisms, and cokernels of monomorphisms. We show some relationships to the theory of prime ideals in the case of noetherian algebras.