Let $S = \oplus\_{d \geq 0} S\_d$ be a graded (Noetherian) ring, let $I \subset S$ be a homogeneous ideal, and let $f \in S$ be a homogeneous element. Denote by $S\_{(f)}$ the subring of degree-$0$ elements of the localization $S\_f$.



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> Do we always have $(H^i\_{IS\_f}(S\_f))\_0 \simeq H^i\_{IS\_f \cap S\_{(f)}}(S\_{(f)})$ as $S\_{(f)}$-modules? That is, is the degree-zero component of local cohomology the same as local cohomology with respect to the degree-zero components?
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More generally, if $S$ is a graded ring and $J \subset S$ a homogeneous ideal, do we have $(H^i\_J(S))\_0 \simeq H^i\_{J\_0}(S\_0)$ as $S\_0$-modules?


The very thorough chapters on graded local cohomology in Brodmann and Sharp's book don't address this, unless I missed it somehow.



Does local cohomology commute with taking the degree-zero component?
The answer to your more general question is indeed positive. This (and its generalisation to $\mathbb{Z}^n$-graduations and to arbitrary degrees) is proven in Lemma 14.1.12 in the second edition of the book by Brodmann and Sharp (*Local cohomology (second edition),* Cambridge University Press, 2013) mentioned in the question.


