# Examples/other referrals for EGA 0.4.5.4

Suggestion 0.4.5.4 in EGA seems a basic representability theory. It reviews:

Intend $F$ is a contravariant functor from the group of in your area ringed rooms over $S$ to the group of collections. Intend offered representable sub-functors $F_i$ of $F$, such that the morphisms $F_i \to F$ are representable by open immersions. Intend in addition that if $Hom(-, X) \to F$ is a morphism and also the functors $F_i \times_F Hom(-,X)$ are representable by $Hom(-,X_i)$, the family members $X_i$ creates an open treatment of $X$. (That $X_i \to X$ is an open immersion adheres to from the interpretations.) Ultimately, intend that if $U$ varies over the open parts of an in your area ringed room $X$, the functor $U \to F(U)$ is a sheaf. After that, $F$ is representable.

I have not yet had the ability to grok the evidence, yet it seems some type of extensive gluing building and construction. This outcome seems made use of in confirming that fibered items exist in the group of systems. Nonetheless, it's rather very easy to straight construct fibered items by gluing open affines.

Exist instances where this outcome in fact makes the life of algebraic geometers less complicated? Additionally, I would certainly value any kind of web links to instances (beyond EGA) where this outcome is made use of.

Actually the outcome is made use of regularly in the essentials of algebraic geometry. It gives *the * formalization of the concept of gluing building and constructions.

As an example if intend to show that fibered items $X \times_S Y$ , first do it for affine systems $X,Y,S$ making use of the adjunction in between $\operatorname{Spec}$ and also international areas. Currently, if $S$ is approximate, 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 also in your area representable on $S$ , hence representable. Hence $X \times_S Y$ exists. Currently if additionally $X$ is approximate, take into consideration 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 also in your area representable on $X$ , hence representable, which reveals that $X \times_S Y$ exists. The common "impromptu" evidence for the presence of the fibered item in fact simply condemn the basic representabilty cause the grandfather clause.

Below is an additional instance : If $A$ is a seemingly - systematic sheaf of algebras on a system $X$ , it is feasible to construct the $X$ - system $Spec(A)$ . It is really tiresome to examine all the information of the gluing building and construction, well - definedness etc when you simply intend to adhesive the $U$ - systems $\operatorname{Spec}(A(U))$ , $U \subseteq X$ affine, with each other. Yet rather, you can simply take into consideration 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 also show that it is a sheaf (noticeable) and also in your area on $X$ representable (common adjunction with range of a ring), so it is representable by an $X$ - system $\operatorname{Spec}(A)$ for which you additionally straight have a global building. Once more I intend to stress : You get involved in a large mess when you intend to construct this without making use of functors or global buildings. These instead abstract ideas are really valuable additionally in concrete scenarios, due to the fact that they make you able to repair your suggestions and also make every building and construction meshed perfectly. When you get even more familiar with these strategies, you stop considering details functors, yet you assume in a "functorial means" and also identify, as an example, why gluing building and constructions job.

I assume one regular scenario when this would certainly be made use of (which you possibly currently recognize). is when you intend to lower the building and construction of some object which is intended to exist over. a base to the instance when the base is affine.

Particularly, visualize you intend to construct a system T over a base S, and also you cover S by open affines S_i ; after that if you construct the systems T_i (pull-back of T over S_i) with ideal gluing information, you will certainly be done.

So, in technique, you will not have T yet will certainly rather have the functor it stands for on S-schemes, T_i will certainly be the fiber item of that functor and also S_i (i.e. simply limit the functor to S_i-schemes), and also currently if T is a sheaf, and also the T_i are representable, you are done.

You can visualize using this to construct projective rooms or Grassmanians over T. E.g. intend you had an in your area free sheaf where you intended to create the linked projective room package. One can do this straight, with some Proj building and construction, yet one can rather identify the global building, and afterwards change S by an open cover S_i over which the in your area free sheaf is in fact free, in which instance the common projective room of the ideal measurement. ( taken control of S_i) will clearly represent your functor (given you figured the functor out appropriately!).

I need to claim that, beyond the context of EGA itself, I do not visualize that individuals mention this outcome (or similar ones) really usually. They are more probable simply to write something like "given that F is a sheaf, we can lower our building and construction to the instance when S is affine". It is simply among the typical strategies that float about for attempting to stand for moduli troubles.