<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.  What I know:

(SOME EDITS below here...)

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

* If $M$ has a pre-dual, say $N^\vee = M$, $\mathbb{V}(N)=Spec(Sym(N))$ does not generally work (thanks to a fortiori for pointing this out)

* EGA I (1971) 9.4.10 mentions in passing, without proof, that this is only possible when $M$ is locally free of finite rank; I'm not sure if this is correct, but if it is, I would like to see the proof.

<b>Why these restrictions on $M,A$?</b>

I want the answer to this question to provide a rigorous geometric intuition about the module $M$.  For example, I want to view $Spec(A/I)\subset Spec(A)$ as the support of the $A$-module $A/I$, which doesn't often have a predual. In practice, I mostly want  this geometric interpretation for finitely generated modules over a Noetherian ring (and at the moment, the partial answer to Dinakar's question doesn't suit my purposes).  So,

<b>Question:</b>

>If $M$ is a finitely generated $A$-module, $A$ Noetherian, under what necessary and sufficient conditions on $M$ is the functor <b>$\otimes_A M: A$-alg $\to$ Set</b> representable by an $A$-scheme?

This may not have a useful answer, or perhaps it has several.  Any help is appreciated!

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