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arXiv · 2003.03688

A new dimension for Grothendieck categories via the atom spectrum

Negar Alipour·Reza Sazeedeh
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Abstract

In this paper, we define a new dimension for objects in a Grothendieck category $\mathcal{A}$. 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 $A$. We introduce a new spectrum Comp$\,A$ via compressible right $A$-modules. In analogy with dimension theory for commutative rings, we show that the Krull dimension of right $A$-modules can be computed via the length of chain of prime ideals of $A$ and also the length of chain of elements of Comp$\,A$.

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Negar Alipour, Reza Sazeedeh. 2026-01-23. A new dimension for Grothendieck categories via the atom spectrum. https://arxiv.org/abs/2003.03688

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