Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $j$ ?

share|cite|improve this question

1 Answer 1

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.