I am looking for a counter example to the fact that a faithfully flat morphism is
an effective descent morphism for the category of quasi-coherent sheaves
when one forgets the quasi-compact hypothesis. If possible, I'd like a counter-example
on the *descent of morphism aspects*. In other words, I am looking for
a morphism of scheme $X' \rightarrow X$, faithfully flat but not quasi-compact,
and two quasi-coherent sheaves $F$ and $G$ on $X$, such that, calling $X''=X' \times\_X X'$,
$F'$, $F''$, $G'$, $G''$ thepull-back of $F$,$G$ to $X'$, $X''$, the sequence of modules
$$0 \rightarrow Hom\_X(F,G) \rightarrow Hom\_{X'}(F',G') \rightarrow Hom\_{X''}(F'',G'')$$
is not exact (here the last morpheme is as it should the difference between the morphism 
induced by the two projections $X'' -> X'$. I wrote it this way because I am not sure how to do a double arrow in mathoverflow's tex:-)



Counter-example to faithfully flat descent
I think the following should be a counter-example.


Let $X=\text{Spec}(\mathbb{C}[T])$ be the affine line and $X'=\bigsqcup\_{x\in\mathbb{C}}\text{Spec}(\mathbb{C}[T]\_{(T-x)})$ be its faithfully flat cover built from all local rings at closed points. Let $F=\mathcal{O}\_{X}$ and $G=\bigoplus\_{x\in\mathbb{C}}\mathcal{O}\_X/(T-x)\simeq\bigoplus\_{x\in\mathbb{C}}\mathbb{C}\_x$ be two quasi-coherent sheaves on $X$.


Then, there is a natural quotient morphism $u':F'\to G'$ given by $\mathbb{C}[T]\_{(T-x)}\to \mathbb{C}[T]/(T-x)$ on $\text{Spec}(\mathbb{C}[T]\_{(T-x)})$. This morphism obviously satisfies the descent condition, as the only non-trivial conditions to check are on connected components of $X''$ that are isomorphic to the generic point of $X$, and $G''$ is zero on these components. However, this morphism does not come from a morphism $u:F\to G$ because $u(1)$ should have a non-zero component in every factor of the direct sum, which is impossible.


