  Let $A$ be a virtually Gorenstein algebra of finite CM-type. We establish a
duality between the subcategory of compact objects in the homotopy category of
Gorenstein projective left $A$-modules and the bounded Gorenstein derived
category of finitely generated right $A$-modules. Let $R$ be a two-sided
noetherian ring such that the subcategory of Gorenstein flat modules
$R\mbox{-}\mathcal{GF}$ is closed under direct products. We show that the
inclusion $K(R\mbox{-}\mathcal{GF})\to K(R\mbox{-}{\rm Mod})$ of homotopy
categories admits a right adjoint. We introduce the notion of Gorenstein
representation dimension for an algebra of finite CM-type, and establish
relations among the dimension of its relative Auslander algebra, Gorenstein
representation dimension, the dimension of the bounded Gorenstein derived
category, and the dimension of the bounded homotopy category of its Gorenstein
projective modules.
