Relative derived dimensions for cotilting modules

For a Noetherian ring $R$ and a cotilting $R$-module $T$ of injective dimension at least $1$, we prove that the derived dimension of $R$ with respect to the category ${\mathcal{X}}_T$ is precisely the injective dimension of $T$ by applying Auslander-Buchweitz theory and Ghost Lemma. In particular, when $R$ is a commutative Noetherian local ring with a canonical module ${\omega }_R$ and $\dim R\ge 1$, the derived dimension of R with respect to the category of maximal Cohen-Macaulay modules is precisely $\dim R$.