  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.
