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In [Hartshorne, III.3] he proves that injective modules over $R$ give flasque sheaves over $Spec\ R$. I presume that's because they don't give injective sheaves, and flasque is the consolation prize. Is there an easy counterexample?

EDIT: in III.3 he's assuming Noetherian. And he's already proved in II.5.5 the equivalence of categories of $R$-modules and quasicoherent ${\mathcal O}_{Spec\ R}$-modules. (And that injective sheaves are flasque, in III.2.)

EDIT: his proof that injectives are flasque uses some non-quasicoherent sheaves. So the ingredients "injective R-modules give injective objects in the category of quasicoherent sheaves [II.5.5]" plus "injective objects in the category of sheaves are flasque [III.2]" isn't enough for the result he gets in III.3, that injective R-modules give flasque sheaves.

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The sheaf is injective in the category of quasi coherent sheaves, though. –  Mariano Suárez-Alvarez Feb 9 '10 at 4:40
@Allen, this is really in Chapter III, I guess? –  Mariano Suárez-Alvarez Feb 9 '10 at 6:21
Oops, fixed II -> III. –  Allen Knutson Feb 9 '10 at 12:55
Good question, I was also wondering about this yesterday. –  Wanderer Feb 9 '10 at 14:27

3 Answers 3

up vote 6 down vote accepted

Let me put this here for the sake of clarity. As was noted by Emerton in a comment above, this answer to a related Math Overflow question shows that the answer is no, for an injective $R$-module $I$, the sheaf $\widetilde{I}$ is not necessarily an injective sheaf.

So if you like that answer, I suggest you click on the link above and upvote that answer. The reference provided in that answer is to the following:

MR0617087 (82i:13013) Dade, Everett C. Localization of injective modules. J. Algebra 69 (1981), no. 2, 416--425.

Localization of modules over a commutative ring $R$ with respect to a multiplicatively closed subset $S$ of $R$ is an exact functor with a large number of properties, some of which are listed in Theorem 3.76 of J. J. Rotman's book [An introduction to homological algebra, Academic Press, New York, 1979; MR0538169 (80k:18001)]. The fifth property, namely: (LI) the localization $S^{-1}E$ of any injective $R$-module $E$ is an injective $S^{-1}R$-module, is false. Two examples are given here showing that arbitrary $R$ and $S$ need not have the property (LI). Also a positive result is given, showing that (LI) holds for certain non-Noetherian $R$ and certain $S$. In particular, if $R$ is the polynomial ring $k[x_1,x_2,\cdots]$ in a countable number of $x_n$ over a nonzero Noetherian ring $k$, then (LI) holds for all choices of $S$.

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In Residues and duality, Corollary 7.14 of Chapter II, he proves that when $R$ is noetherian the sheafification of an injective module is an injective $\mathcal O_X$-module.

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Of course, there exists non-noetherian rings... Hic sunt dracones. –  Mariano Suárez-Alvarez Feb 9 '10 at 5:15
See also this answer: mathoverflow.net/questions/6762/… –  VA. Feb 9 '10 at 5:15
Just to clarify, it's Corollary 7.14 of Chapter II. This is a pretty surprising result to me. It means that the counterexample Allen was asking for probably requires you to think of an example of an injective module I over a non-noetherian ring R, a non-quasi-coherent ("incoherent"?) sheaf F on Spec(R), and a non-split injection I→F. Injective modules, non-noetherian rings, and incoherent sheaves are pretty hard for me to think about individually; putting them all together may be hopeless. –  Anton Geraschenko Feb 9 '10 at 6:04
Added the chapter, which is always nice to have. –  Mariano Suárez-Alvarez Feb 9 '10 at 6:21
Adam Topaz's answer to the question linked to by VA gives a counterexample (although I didn't check the reference). –  Emerton Feb 9 '10 at 20:29

I wonder why no one has mentioned this yet: A counterexample by Verdier can be found in SGA 6, Exp. 2, App. I. You can read it here. It is also shown that the forgetful functor $\mathrm{Qcoh}(X) \to \mathrm{Mod}(X)$ is ill-behaved on derived categories. Another example can be found in the Stacks Project's Examples (#26).

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I like this appendix by Verdier. Thanks for putting it here. –  Mahdi Majidi-Zolbanin Dec 29 '12 at 22:02

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