Finiteness of the number of minimal atoms in Grothendieck categories

TL;DR

This paper proves that Grothendieck categories with a noetherian generator have finitely many minimal atoms, paralleling the finiteness of irreducible components in noetherian schemes, and shows each minimal atom corresponds to a compressible object.

Contribution

It establishes the finiteness of minimal atoms in Grothendieck categories with noetherian generators and links minimal atoms to compressible objects, extending classical geometric results.

Findings

Finitely many minimal atoms in such categories

Each minimal atom is represented by a compressible object

Analogy with irreducible components in schemes

Abstract

For a Grothendieck category having a noetherian generator, we prove that there are only finitely many minimal atoms. This is a noncommutative analogue of the fact that every noetherian scheme has only finitely many irreducible components. It is also shown that each minimal atom is represented by a compressible object.