Homological Dimensions of Extriangulated Categories and Recollements

TL;DR

This paper introduces and analyzes the concepts of global and extension dimensions within extriangulated categories, providing bounds and generalizations that unify results from abelian and triangulated category recollements.

Contribution

It defines new homological dimensions for extriangulated categories and establishes bounds that extend known results from related categorical frameworks.

Findings

Upper bounds for global dimensions in extriangulated recollements

Upper bounds for extension dimensions in extriangulated recollements

Unified generalization of results from abelian and triangulated categories

Abstract

In this paper, let $(\mathcal{A},\mathcal{B},\mathcal{C})$ be a recollement of extriangulated categories. We introduce the global dimension and extension dimension of extriangulated categories, and give some upper bounds of global dimensions (resp. extension dimensions) of the categories involved in $(\mathcal{A},\mathcal{B},\mathcal{C})$, which give a simultaneous generalization of some results in the recollement of abelian categories and triangulated categories.