Cohomological dimension of complexes

In the derived category of the category of modules over a commutative Noetherian ring $R$, we define, for an ideal $\text{fa}$ 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$\cd(\fa, M)=\sup{\ell\in\Bbb Z|\H^{\ell}{\fa}(M)\neq 0}$foran$R$--module$M$\mathrm{.}Inthispaper,itisshown,amongotherthings,that,foranycomplex$X$boundedtotheleft,$-\inf {\mathbf R}\G{\fa}(X)\le\cd(\fa, X)$ and equality holds if indeed $\stackrel{˝}{(}X)$ is finitely generated.