  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.
