I'll start with example:


Let $X$ be a scheme, and $O\_X$ be its structure sheaf. It is defined at the moment on open sets of $X$, and it takes them to $Sets$. However, it is extendable to a sheaf on the Zariski site of $Sch$ by: Take a scheme $S$ to $\mathbb{G}\_a(S)$. Now that it is a functor $Sch \rightarrow Sets$, it makes sense to ask whether it is representable, which in this case it is (by $\mathbb{Z}[X]$), and it is even a group scheme.


My, somewhat vague, question is: how prevalent is the phenomenon? For example, are all coherent sheaves on any scheme extendable to representable functors? To group schemes? Is there an iff condition for this to happen?



When is a sheaf on a scheme extendable to a representable functor?
If $F$ is a coherent sheaf on a noetherian scheme $X$, there is a natural extension of $F$ to the large Zariski site of $X$: with an object $f\colon T \to X$, you associate the group of global sections of the pullback $f^\*F$. According to a result of Nitin Nitsure, this is representable if and only if $F$ is locally free (see <http://arxiv.org/abs/math/0308036>). What Keerthi says is not quite correct: the functor represented by the spectrum of the symmetric algebra of $F$ is that sending $f\colon T \to X$ to the group of global sections of the dual of $f^\*F$, which does not coincide with the group of global sections of $f^\*(F^\vee)$.


On the other hand there are many ways of extending a sheaf on the small Zariski site; for example, one can extend it to the small étale site, where it is always represented by an algebraic space with an étale map to $X$ (the analogue of the "espace étalé" for the usual topology), which then you can extend to the large étale site. This would have a somewhat better chance of being representable by a scheme; however, this construction is very different in spirit, and the resulting scheme would be enormous, and probably not very useful.


