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The study of the cohomological dimension of algebraic varieties
has produced some interesting results and problems in local
algebra. Let $\frak{a}$ be an ideal of a commutative Noetherian
ring $R$. For finitely generated $R$-modules $M$ and $N$, the
concept of cohomological dimension $cd_{\frak {a}}(M,N)$ of $M$
and $N$ with respect to $\frak{a}$ is introduced. If $0\rightarrow
N' \rightarrow N'' \rightarrow 0$ is an exact sequence of finitely
generated $R$-modules, then it is shown that $cd_{\frak {a}}(M,N)$
= max$ \{{cd_{\frak{a}}(M,N'),cd_{\frak{a}}(M,N'')}\}$ whenever
proj dim $ M <\infty$. Also, if $L$ is a finitely generated
$R$-module with Supp$(N/\Gamma_{\frak{a}}(N)) \subseteq$ Supp
$(L/\Gamma_{\frak{a}}(L))$, then it is proved that $cd_{\frak
{a}}(M,N) \leq$ max\{${cd_{\frak{a}}(M,L),proj\ dim M}\}$.
Finally, as a consequence, a result of Brodmann is improved.
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