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I am trying to verify that:

A sheaf of $\mathcal{O}_X$ modules $\mathcal{F}$ is an injective object in the category of $\mathcal{O}_X$ modules iff its local rings $\mathcal{F}_x$ are injective $\mathcal{O}_x$ modules for each $x\in X$.

I have a proof(??) to say that this is actually true but it is too long to write here. I would write here if it is false, then some one can point out the wrong justification that I have given. If it is true and well known it is of not much use to write down here.

If not for any scheme $X$ is it true for Noetherian schemes atleast? Any suggestion is welcome.

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    $\begingroup$ By Proposition II.7.17 in Hartshorne's Residues and duality, this holds if $X$ is locally noetherian and $\mathcal{F}$ is quasicoherent. $\endgroup$
    – Fred Rohrer
    Commented Sep 8, 2017 at 21:20
  • $\begingroup$ @FredRohrer Thanks for your comment. I have not yet seen that book as I have started to understand sheaf cohomology (which is where I had to use this result) only recently. I think this is the best reason to look at that book at least if not to really understand the proof. Thank you. $\endgroup$
    – user37663
    Commented Sep 9, 2017 at 2:56
  • $\begingroup$ @FredRohrer I was giving some wrong justification which I have realised. Lemma 7.12 says that even for one direction it needs the assumption that $X$ is locally noetherian. It says the following: Let $X$ be a locally Noetherian prsescheme, and let $\mathcal{I}$ be an injective $\mathcal{O}_X$ module. Then the stalk $\mathcal{I}_x$ of $\mathcal{I}$ at each point $x\in X$ is an injective module over the local ring $\mathcal{O}_x$ of $x$. Your comment is very useful. Thanks. $\endgroup$
    – user37663
    Commented Sep 9, 2017 at 6:38
  • $\begingroup$ Dear Praphulla, the result you cite does not say that it needs local noetherianness. However, I suspect that this hypothesis can indeed not be omitted (since, for modules over rings, localisation does not necessarily preserve injectivity in a non-noetherian situation, cf. work of Dade). $\endgroup$
    – Fred Rohrer
    Commented Sep 9, 2017 at 6:50
  • $\begingroup$ @FredRohrer : :) By "... needs .." I mean they have used the condition of local noetherian. Yes, I do remember seeing something which says there are Injective modules over a non noetherian ring such that their localizations are not injective modules over repsective localization of rings. Yes, I have heard about the paper by Everett C Dade which is online here sciencedirect.com/science/article/pii/0021869381902131 $\endgroup$
    – user37663
    Commented Sep 9, 2017 at 7:02

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