Recollements of Cohen-Macaulay Auslander algebras and Gorenstein derived categories

Let $A$, $B$ and $C$ be associative rings with identity. Using a result of Koenig we show that if we have a $\mathbb{D}^{{\rm{b}}}({\rm{{mod\mbox{-}}}} )$ level recollement, writing $A$ in terms of $B$ and $C$, then we get a $\mathbb{D}^-({\rm{Mod\mbox{-}}} )$ level recollement of certain functor categories, induces from the module categories of $A$, $B$ and $C$. As an application, we generalise the main theorem of Pan [Sh. Pan, Derived equivalences for Cohen-Macaulay Auslander algebras, J. Pure Appl. Algebra, 216 (2012), 355-363] in terms of recollements of Gorenstein artin algebras. Moreover, we show that being Gorenstein as well as being of finite Cohen-Macaulay type, are invariants with respect to $\mathbb{D}^{{\rm{b}}}_{{{\mathcal{G}p}}}({\rm{{mod\mbox{-}}}})$ level recollements of virtually Gorenstein algebras, where $\mathbb{D}^{{\rm{b}}}_{{{\mathcal{G}p}}}$ denotes the Gorenstein derived category.