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 XT is precisely the injective dimension of T by applying Auslander-Buchweitz theory and Ghost Lemma. In particular, when R is a commutative Noetherian Cohen-Macaulay local ring with a canonical module ωR and dim⁡R≥1, the derived dimension of R with respect to the category of maximal Cohen-Macaulay modules is precisely dim⁡R.