is the property of representability of a sheaf on the big etale site checkable on the small site?

Question

Let $S$ be a scheme and $F$ a sheaf on $(\textbf{Sch}/S)_\text{etale}$, whose restriction to the small etale site $S_\text{etale}$ is representable (in fact in my case this restriction is representable by a scheme finite etale over $S$). Must $F$ be representable?

Answer 1

It's not true. Basically what goes wrong is that there is no way to control what happens over objects in $\mathbf{Sch}_{/ S}$ that are "far" from being étale over $S$.

Let $S = \operatorname{Spec} K$ for a field $K$ and let $L$ be a field extension of $K$ such that $\dim_K L$ is uncountable. If $A$ is a finitely generated $K$-algebra, then $\dim_K A$ is (finite or) countable, so any $K$-algebra homomorphism $L \to A$ must have $A \cong \{ 0 \}$.

Let $T = \operatorname{Spec} L$ and let $F$ be any non-representable étale sheaf on $\mathbf{Sch}_{/ S}$ with a morphism $F \to h_T$. The small étale site over $S$ is generated by the finite étale algebras over $K$, so the above argument shows that the restriction of $h_T$ to the small étale site is isomorphic to the sheaf represented by $\emptyset = \operatorname{Spec} \{ 0 \}$. But that implies that the restriction of $F$ to the small étale site is also isomorphic to the sheaf represented by $\emptyset$, so we have the desired counterexample.

To be explicit, I suppose we could take something like $F = \mathop{{\varinjlim}_n} {\mathbb{A}^n_T}$, where the colimit is taken in the category of étale sheaves.