Foxby equivalence, local duality and Gorenstein homological dimensions

Let $(R,\text{fm})$ 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 $\text{fm}$-section complex ${\bf R}\Gamma_{\fm}({\bf R}\Hom_R(X,Y))$\mathrm{.}Inparticular,ifG-dimensionof$X$ and injective dimension of $Y$ are finite, then we show that $$\mathbf{R}{\mathrm{\Gamma }}_{\text{fm}}(\mathbf{R}{\text{Hom}}_R(X,Y))\simeq (\mathbf{R}{\text{Hom}}_R(Y,D{\otimes }_R^{\mathbf{L}}X){)}^{\vee }\mathrm{.}$$ 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.