Can injective modules over R give non-injective sheaves over Spec R?

Question

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.

Answer 1

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 $S$.

Answer 2

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.

Answer 3

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).