Let $F: \mathsf{Sch_{/S}}^{op} \to \mathsf{Set}$ be a Zariski sheaf on the category of $S$-schemes. $F$ being a sheaf means it satisfies the following property:
Sheaf condition: For every $S$-scheme $X$ and every open cover $\{U_j\} \subset Open_S(X)$ of $X$ by open $S$-subschemes. The following diagram is an equalizer:
$$F(X) \rightarrow \prod_{i} F(U_i) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{i, j} F(U_i \times_X U_j)$$
The following theorem gives a necessary and sufficient condition for $F$ to be representable by an $S$-Scheme:
Theorem: $F$ is representable by an $S$-scheme iff $F$ has an open cover by representable subfunctors.
This is pretty satisfying, but let's try something even bolder.
Is representability local on the base? : Let $F: \mathsf{Sch_{/S}}^{op} \to \mathsf{Set}$ be a Zariski sheaf and $\{U_j\} \subset Open(S)$ an open cover of $S$ satisfying that all pullbacks $F \times_S U_j$ are representable. Must $F$ be representable?
The just glue attitude doesn't seem to work for me here. I've played for hours with pullback cubes and the like without getting anywhere.
Unless I'm misunderstanding something (which I probably do) this version of "locality" is used implicitly in a lot of arguments I've encountered. Is it really true? If so, why?
If not, why not and is there perhaps a different locality principle which does hold?
Perhaps I'm missing something, but isn't this just the classical case of gluing schemes along open subschemes?
Since $F\times_S U_j$ is representable, say by a scheme $X_j$ over $U_j$, then for any $i,j$, $X_j|_{U_i\cap U_j}$ is an open subscheme of $X_j$, and $X_i|_{U_i\cap U_j}$ is an open subscheme of $X_i$. Since they're both pullbacks of $F$ along the same morphism $U_i\cap U_j\rightarrow S$, they are "uniquely" isomorphic, and the associated isomorphisms for triples of indices $i,j,k$ must satisfy the cocycle condition (again by uniqueness of isomorphisms), so now you have a family of schemes $X_i$ and various open subschemes and isomorphisms between the open subschemes, so you're in the classical gluing situation (see, for example, Hartshorne chapter II, Exercise 2.12), from which you can construct a scheme $X$ over $S$, whose functor of points must coincide with $F$ because of "the fully faithfulness of pullback functors of descent data".