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Fix $r \\geq 1$. If $J$ is a homogeneous ideal of $G_{P^r}(A)$ with $\\text{dim} \\ G_{P^r}(A)/J > 0$ then the local cohomology module $H^d_J(G_{P^r}(A)) = 0$.\n  (2) Let $A = K[[X_1, \\ldots,X_d]]$ and let $\\mathfrak{m} = (X_1, \\ldots, X_d)$. Assume $K$ is separably closed. Fix $r \\geq 1$. Let $J$ be a homogeneous ideal of $G_{\\mathfrak{m}^r}(A)$. We show that local cohomology modules $H^{j}_J(G_{\\mathfrak{m}^r}(A)) = 0$ for $j \\geq d ","authors_text":"Tony J. 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