Recently, I came across this answer on MO, which (allegedly - I have trouble understanding EGA/SGA, so I have not checked the reference) provides a reference for an injective $A$-module, $A$ a commutative ring, together with an injective $A$-module $I$, such that $\tilde{I}$, i.e. the $\mathrm{Spec}A$-module defined by localization of $I$ on the principal open sets, is not injective as a quasi coherent sheaf.
My question is:
How come? More precisely, what is wrong with the following argument?
Set $X:=\mathrm{Spec}A$. I would use the following two facts:
1) Any quasi-coherent $\mathscr{O}_X$-module $\mathscr{F}$ is of the form $\tilde{M}$ for some $M \in A-\mathrm{Mod}$ - this (I think) is roughly what Theorem 17.6(i) in Görtz's & Wedhorn's Algebraic geometry I says, since $\mathrm{Spec}A$ is affine.
2) The functor $M \mapsto \tilde{M}$ is fully faithful (again Görtz-Wedhorn, Proposition 7.13).
Since 1) implies that the global section functor is inverse to the one from 2), this gives me the equivalence of categories $Qcoh_X$ and $\mathrm{Mod}-A$. In particular, injective objects are mapped to injective objects and vice versa. Or am I missing something?
Thank you for pointing out mistakes in the reasoning. I would also appreciate any references for similar counterexample in English (provided that the argument is false and the counterexample exists).
(I would also like to apologize for the title which sounds terrible to me, but I dod not know how else to describe the situation briefly.)
What Verdier actually shows is that there exists an affine scheme $X = \mathrm{Spec}(A)$ and an injective $A$-module $I$ such that $\tilde{I}$ is not an injective $\mathscr{O}_X$-module. The latter is of course defined as an injective object of $\mathrm{Mod}(X)$, which is not the same thing as an injective object of $\mathrm{QCoh}(X)$. As Kevin Carlson remarked, if $A$ is noetherian then $\tilde{I}$ will be injective as an $\mathscr{O}_X$-module (see [Hartshorne, Residues and duality, II, Corollary 7.14]).