The derived and extension dimensions of abelian categories

TL;DR

This paper explores the relationship between derived and extension dimensions in abelian categories, providing new bounds for these dimensions in the context of artin algebras based on algebraic properties.

Contribution

It establishes a connection between derived and extension dimensions and introduces new upper bounds for these dimensions for artin algebras.

Findings

Derived and extension dimensions are related in abelian categories.

New upper bounds for extension dimension of artin algebras are provided.

Upper bounds for derived dimension are derived from algebraic properties.

Abstract

For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $\Lambda$, we give the upper bounds of the extension dimension of $\Lambda$ in terms of the radical layer length of $\Lambda$ and certain relative projective (or injective) dimension of some simple $\Lambda$-modules, from which some new upper bounds of the derived dimension of $\Lambda$ are induced.