A new dimension for Grothendieck categories via the atom spectrum
Category Theory
2026-01-26 v9
Abstract
In this paper, we define a new dimension for objects in a Grothendieck category . We show that it serves as a lower bound for Gabriel-Krull dimension and under certain conditions, the two dimensions coincide. We carry out our investigation for a fully right bounded ring . We introduce a new spectrum Comp via compressible right -modules. In analogy with dimension theory for commutative rings, we show that the Krull dimension of right -modules can be computed via the length of chain of prime ideals of and also the length of chain of elements of Comp.
Cite
@article{arxiv.2003.03688,
title = {A new dimension for Grothendieck categories via the atom spectrum},
author = {Negar Alipour and Reza Sazeedeh},
journal= {arXiv preprint arXiv:2003.03688},
year = {2026}
}