# The Extension Dimension of Abelian Categories

## TL;DR

This paper explores the extension dimension in abelian categories with enough projectives and injectives, establishing bounds related to representation dimension and analyzing its behavior under ring extensions.

## Contribution

It proves the equality of extension and weak resolution dimensions under certain conditions and relates them to the representation dimension, providing bounds and behavior analysis.

## Key findings

- Extension dimension equals weak resolution dimension in certain abelian categories.
- Provides upper bounds for extension dimension based on projective dimensions and radical layer length.
- Analyzes the behavior of extension dimension under ring extensions and recollements.

## Abstract

Let $\A$ be an abelian category having enough projective objects and enough injective objects. We prove that if $\A$ admits an additive generating object, then the extension dimension and the weak resolution dimension of $\A$ are identical, and they are at most the representation dimension of $\A$ minus two. By using it, for a right Morita ring $\La$, we establish the relation between the extension dimension of the category $\mod \La$ of finitely generated right $\Lambda$-modules and the representation dimension as well as the right global dimension of $\Lambda$. In particular, we give an upper bound for the extension dimension of $\mod \Lambda$ in terms of the projective dimension of certain class of simple right $\Lambda$-modules and the radical layer length of $\Lambda$. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements.