Examples/other references for EGA 0.4.5.4

Proposition 0.4.5.4 in EGA appears to be a general representability theorem. It reads:

Suppose $F$ is a contravariant functor from the category of locally ringed spaces over $S$ to the category of sets. Suppose given representable sub-functors $F_i$ of $F$, such that the morphisms $F_i \to F$ are representable by open immersions. Suppose furthermore that if $Hom(-, X) \to F$ is a morphism and the functors $F_i \times_F Hom(-,X)$ are representable by $Hom(-,X_i)$, the family $X_i$ forms an open covering of $X$. (That $X_i \to X$ is an open immersion follows from the definitions.) Finally, suppose that if $U$ ranges over the open subsets of a locally ringed space $X$, the functor $U \to F(U)$ is a sheaf. Then, $F$ is representable.

I haven't yet been able to grok the proof, but it appears to be some sort of extended gluing construction. This result appears to be used in proving that fibered products exist in the category of schemes. However, it's fairly easy to directly construct fibered products by gluing open affines.

Are there examples where this result actually makes the life of algebraic geometers easier? Also, I'd appreciate any links to examples (outside of EGA) where this result is used.

I think one typical situation when this would be used (which you probably already know) is when you want to reduce the construction of some object which is supposed to lie over a base to the case when the base is affine.

Namely, imagine you want to build a scheme T over a base S, and you cover S by open affines S_i; then if you construct the schemes T_i (pull-back of T over S_i) with appropriate gluing data, you will be done.

So, in practice, you won't have T but will instead have the functor it represents on S-schemes, T_i will be the fibre product of that functor and S_i (i.e. just restrict the functor to S_i-schemes), and now if T is a sheaf, and the T_i are representable, you are done.

You could imagine applying this to build projective spaces or Grassmanians over T. E.g. suppose you had a locally free sheaf from which you wanted to form the associated projective space bundle. One could do this directly, with some Proj construction, but one could instead figure out the universal property, and then replace S by an open cover S_i over which the locally free sheaf is actually free, in which case the usual projective space of the appropriate dimension (taken over S_i) will obviously represent your functor (provided you figured the functor out correctly!).

I should say that, outside of the context of EGA itself, I don't imagine that people cite this result (or analogous ones) very often. They are more likely just to write something like "since F is a sheaf, we can reduce our construction to the case when S is affine". It is just one of the standard techniques that float around for trying to represent moduli problems.

Actually the result is used all the time in the basics of algebraic geometry. It provides the formalization of the principle of gluing constructions.

For example if want to show that fibered products $X \times_S Y$, first do it for affine schemes $X,Y,S$ using the adjunction between $\operatorname{Spec}$ and global sections. Now, if $S$ is arbitrary, the functor $\operatorname{Sch}/S \to \operatorname{Set}, Z \mapsto \operatorname{Hom}_S(Z,X) \times \operatorname{Hom}_S(Z,Y)$ is a sheaf and locally representable on $S$, thus representable. Thus $X \times_S Y$ exists. Now if also $X$ is arbitrary, consider the functor $\operatorname{Sch}/X \to \operatorname{Set}, Z \mapsto \operatorname{Hom}_S(Z',Y)$, where $Z' = Z \to X \to S$. This is a sheaf and locally representable on $X$, thus representable, which shows that $X \times_S Y$ exists. The usual "ad hoc" proofs for the existence of the fibered product actually just reprove the general representabilty result in the special case.

Here is another example: If $A$ is a quasi-coherent sheaf of algebras on a scheme $X$, it is possible to construct the $X$-scheme $Spec(A)$. It is very laborious to check all the details of the gluing construction, well-definedness etc. when you just want to glue the $U$-schemes $\operatorname{Spec}(A(U))$, $U \subseteq X$ affine, together. But instead, you could just consider the functor

$$\operatorname{Sch}/X \to \operatorname{Set}, (t : Z \to X) \mapsto \operatorname{Hom}_{\mathcal{O}_X-\operatorname{Alg}}(A,t_* \mathcal{O}_Z)$$

and show that it is a sheaf (obvious) and locally on $X$ representable (usual adjunction with spectrum of a ring), so it is representable by an $X$-scheme $\operatorname{Spec}(A)$ for which you also directly have a universal property. Again I want to emphasize: You get into a big mess when you want to construct this without using functors or universal properties. These rather abstract notions are very useful also in concrete situations, because they make you able to fix your ideas and make every construction fit together nicely. When you get more accustomed to these techniques, you stop thinking of specific functors, but you think in a "functorial way" and recognize, for example, why gluing constructions work.