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?

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    $\begingroup$ The dual of any coherent sheaf is represented by the relative Spec of its symmetric algebra: in particular, any reflexive coherent sheaf is representable by what is tautologically a group scheme. Representability in general is quite subtle, though there are general criteria (due to Artin) for representability by algebraic spaces. I would recommend reading Mumford's 'Lectures on Curves...' (where he talks about representability of the Hilbert scheme) and Kleiman's article on the Picard variety in 'Fundamental Algebraic Geometry: FGA explained'. $\endgroup$ – Keerthi Madapusi Pera Jun 18 '11 at 0:14
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    $\begingroup$ Btw, it was Grothendieck who first brought representability to the forefront. $\endgroup$ – Keerthi Madapusi Pera Jun 18 '11 at 0:14

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.

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  • $\begingroup$ Thanks for the correction and the reference to Nitsure's paper! It looks very interesting. $\endgroup$ – Keerthi Madapusi Pera Jun 18 '11 at 3:58
  • $\begingroup$ The fact that the functor $T\mapsto \Gamma(T,f^\ast F)$ is representable if and only if $F$ is locally free, is already in EGA1, Grundlehren. $\endgroup$ – euklid345 May 3 '16 at 9:07

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