Foxby equivalence, local duality and Gorenstein homological dimensions

Let $(R,\fm)$ be a local ring and $(-)^{\vee}$ denote the Matlis duality functor. We investigate the relationship between Foxby equivalence and local duality through generalized local cohomology modules. Assume that $R$ possesses a normalized dualizing complex $D$ and $X$ and $Y$ are two homologically bounded complexes of $R$-modules with finitely generated homology modules. We present several duality results for $\fm$-section complex ${\bf R}\Gamma_{\fm}({\bf R}\Hom_R(X,Y))$. In particular, if G-dimension of $X$ and injective dimension of $Y$ are finite, then we show that $${\bf R}\Gamma_{\fm}({\bf R}\Hom_R(X,Y))\simeq ({\bf R}\Hom_R(Y,D\otimes_ R^{{\bf L}}X))^{\vee}.$$ We deduce several applications of these duality results. In particular, we establish Grothendieck's non-vanishing Theorem in the context of generalized local cohomology modules.