<b>Motivation:</b>

It's nice when you can think of the elements of an $A$-module $M$ as sections some $A$-scheme <br>$Y\to Spec(A)$.  That is, maps $Spec(A)\to Y$ such that $Spec(A)\to Y \to Spec(A)$ is the identity.

<b>What's wrong with the ["espace étalé"][1]:</b>

One way to do this is to consider the associated sheaf $\tilde{M}$, and form its "espace étalé" $\acute{E}t(\tilde{M})$.  Observe that this topological space is naturally an $X$-scheme (essentially by its construction, as for $\acute{E}t$ of any sheaf of sets), and that $\Gamma(U,\acute{E}t(\tilde{M})) = \tilde{M}(U)$ for opens $U\subseteq Spec(A)$.  

I'm not happy with this construction in that it has "the wrong fibres": for $I\triangleleft A$, the sections of $\acute{E}t(\tilde{M})$ over (base changed to) a closed subscheme $Z(I)$, e.g. a point, do not correspond to $\widetilde{M/IM}$.  This is just an instance of the fact that $\acute{E}t$ <b>doesn't respect base change</b>: given $f:Spec(B)\to Spec(A)$, in general $\acute{E}t(f^* \tilde{M})\neq f^* \acute{E}t(\tilde{M})$. 

<b>Conclusion:</b>

I want a construction that *does* respect base change.  That is, for any module $M$ on $X$, I want an $X$-scheme $Y$ such that for any $X'\to X$, $\Gamma(X',Y_{X'}) = \tilde{M}_{X'}(X')$.  This amounts to representing the functor $B\mapsto B\otimes_A M$ from $A$-algebras to sets.  

<B>The question:</b> (updated, thanks to some comments from a fortiori and buzzard)

>EGA I (1971) 9.4.10 mentions in passing, without proof, that this functor is representable if and only if $M$ is locally free of finite rank.  

>* If this is correct, does anyone know where to find the proof?  

>* If not, does anyone know a correct (and useful) equivalent condition on $M$?

So far, I gather that:

* It is <b>not always representable</b> if $M$ is not finitely generated; [see this earlier question][2].

* If $M$ has a pre-dual, say $N^\vee = M$, $\mathbb{V}(N)=Spec(Sym(N))$ <b>does not generally work</b> (see a fortiori's comment below)

(This may not have a useful answer, or perhaps it has several...)

  [1]: http://en.wikipedia.org/wiki/Sheaf_(mathematics)#The_.C3.A9tale_space_of_a_sheaf
  [2]: http://mathoverflow.net/questions/6764