Cohomological dimension of complexes

In the derived category of the category of modules over a commutative Noetherian ring $R$, we define, for an ideal $\fa$ of $R$, two different types of cohomological dimensions of a complex $X$ in a certain subcategory of the derived category, namely \cd(\fa, X)=\sup\{\cd(\fa, \H_{\ell}(X))-\ell|\ell\in\Bbb Z\} and $-\inf{\mathbf R}\G_{\fa}(X)$, where \cd(\fa, M)=\sup\{\ell\in\Bbb Z|\H^{\ell}_{\fa}(M)\neq 0\} for an $R$--module $M$. In this paper, it is shown, among other things, that, for any complex $X$ bounded to the left, $-\inf {\mathbf R}\G_{\fa}(X)\le\cd(\fa, X)$ and equality holds if indeed $\H(X)$ is finitely generated.