most recent 30 from http://mathoverflow.net 2013-05-19T17:23:44Z http://mathoverflow.net/feeds/question/115726 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115726/artinian-property-of-local-cohomology-module-over-graded-local-ring Axy 2012-12-07T16:42:01Z 2012-12-07T17:19:05Z

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

http://mathoverflow.net/questions/115726/artinian-property-of-local-cohomology-module-over-graded-local-ring/115729#115729 Karl Schwede 2012-12-07T17:19:05Z 2012-12-07T17:19:05Z

<p>Yes, this even holds for non-graded rings. Indeed, suppose that $R$ is a (Noetherian?) ring and $m \subseteq R$ is a maximal ideal. </p> <p>Then $$ H^i_m(M) = H^i_{mR_m}(M_m) $$ essentially by Chapter III, Exercise 2.3(f) (excision) of Hartshorne. </p> <p>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. </p>