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Let $R$ be a Noetherian ring and let $M$ is an $R$-module. Consider the associated affine scheme $(\text{Spec R},\mathcal{O}_{\text{Spec R}})$ and Suppose $Z\subset X$ is a closed subset of $\text{Spec R}$. Is the following is true?

$\Gamma_Z(\widetilde M)\cong\widetilde{ \Gamma_Z(M)}$.

Any information is useful even if it is true in the case that $M$ is an injective $R$-module.

$\widetilde\quad$ is the functor that associates $M$ to the quasicoherent sheaf $\widetilde M$.

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  • $\begingroup$ This is immediate from quasi-coherence of the pushforward to $X$ of the restriction to the quasi-compact open $X-Z$, where $\Gamma_Z$ on the left is "sheaf of sections supported in $Z$"; see Cor. 4, Exp. II, SGA2 for the cohomological generalization for arbitrary rings $R$ and $Z$ for which $X-Z$ is quasi-compact. $\endgroup$
    – user27920
    Commented Aug 9, 2014 at 10:05
  • $\begingroup$ By the way, I think in the question you want to say $\underline{\Gamma}_Z(\widetilde{M})\cong\widetilde{\Gamma_Z(M)}$, because $\Gamma_Z(\widetilde{M})$ is not a sheaf, it's the same thing as $\Gamma_Z(M)$. $\endgroup$
    – Mahdi Majidi-Zolbanin
    Commented Aug 13, 2014 at 23:53

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