Representable Functors

Abstract

In Chapter 4 we attached to a scheme X a contravariant functor h _X from the category of schemes to the category of sets. The Yoneda Lemma 4.6 tells us that we obtain an embedding of the category of schemes into the category of such functors and thus we can consider schemes also as functors. Functors F that lie in the essential image of this embedding are called representable. We say that a scheme X represents F if $h_X \cong F$. It is one of the central problems within algebraic geometry to study functors that classify certain interesting objects and to decide whether they are representable, i.e., whether they are “geometric objects”. For general functors F and G it may be difficult to envisage them as geometric objects. But it makes sense to say that a morphism f : F → G is “geometric” (called representable), even if F and G are not necessarily representable. Thus we may speak of immersions or of open coverings of functors. We will show that a functor that is a sheaf for the Zariski topology and has an open covering by representable functors is itself representable.