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We know that if $(R, m)$ is a local ring, $M$ is a finitely generated $R$-module, then the local cohomology module $H^{i}_{m}(M)$ is an Artinian module for every $j$.

My question is : if $(R,m)$ is a graded local ring, then is $H^{i}_{m}(M)$ is an Artinian module for every $j$ ?

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Yes, this even holds for non-graded rings. Indeed, suppose that $R$ is a (Noetherian?) ring and $m \subseteq R$ is a maximal ideal.

Then $$ H^i_m(M) = H^i_{mR_m}(M_m) $$ essentially by Chapter III, Exercise 2.3(f) (excision) of Hartshorne.

Also note that $N \subseteq H^i_{mR_m}(M_m)$ is an $R$-submodule if and only if it is an $R_m$-module.

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