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In this article, it is shown that there is a finitely generated submodule $N_i$ of $H_{\\frak a}^i(M)$ such that $\\dim{\\rm Supp } H_{\\frak a}^i(M)/N_i<n$ for all $i<t$ if and only if there is a finitely generated submodule $N_{i,{\\frak p}}$ of $H_{{\\frak a} R_{\\frak p}}^i(M_{\\frak p})$ such that $\\dim{\\rm Supp } H_{{\\frak a} R_{\\frak p}}^i(M_{\\frak p})/N_{i,{\\frak p}}<n$ for all $i<t$. 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