Categorical resolutions of a class of derived categories

By using the relative derived categories, we prove that if an Artin algebra $A$ has a module $T$ with ${\rm inj.dim}T<\infty$ such that $^\perp T$ is finite, then the bounded derived category $D^b(A\mbox{-}{\rm mod})$ admits a categorical resolution in the sense of [Kuz], and a categorical desingularization in the sense of [BO]. For CM-finite Gorenstein algebra, such a categorical resolution is weakly crepant. The similar results hold also for $D^b(A\mbox{-}{\rm Mod})$.