Generalized local cohomology and the Intersection Theorem

Let $R$ be commutative Noetherian ring and let $\text{fa}$ be an ideal of $R$. For complexes $X$ and $Y$ of $R$–modules we investigate the invariant $\inf \mathbf{R}{\mathrm{\Gamma }}_{\text{fa}}(\mathbf{R}{\text{Hom}}_R(X,Y))$ in certain cases. It is shown that, for bounded complexes $X$ and $Y$ with finite homology, $\dim Y\le\dim{\mathbf R}\Hom_R(X,Y)\le\pd X+\dim(X\otimes^{\mathbf L}_RY)+\sup X$whichstrengthentheIntersectionTheorem\mathrm{.}Here$\inf X$and$\sup X$ denote the homological infimum, and supremum of the complex $X$, respectively.