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Concept
Functor represented by a scheme
Résumé
In algebraic geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme S is (up to natural bijections) the set of all morphisms . The scheme X is then said to represent the functor and that classify geometric objects over S given by F.
The best known example is the Hilbert scheme of a scheme X (over some fixed base scheme), which, when it exists, represents a functor sending a scheme S to a flat family of closed subschemes of .
In some applications, it may not be possible to find a scheme that represents a given functor. This led to the notion of a stack, which is not quite a functor but can still be treated as if it were a geometric space. (A Hilbert scheme is a scheme, but not a stack because, very roughly speaking, deformation theory is simpler for closed schemes.)
Some moduli problems are solved by giving formal solutions (as opposed to polynomial algebraic solutions) and in that case, the resulting functor is represented by a formal scheme. Such a formal scheme is then said to be algebraizable if there is another scheme that can represent the same functor, up to some isomorphisms.
rational point#Definition
The notion is an analog of a classifying space in algebraic topology. In algebraic topology, the basic fact is that each principal G-bundle over a space S is (up to natural isomorphisms) the pullback of a universal bundle along some map from S to . In other words, to give a principal G-bundle over a space S is the same as to give a map (called a classifying map) from a space S to the classifying space of G.
A similar phenomenon in algebraic geometry is given by a linear system: to give a morphism from a projective variety to a projective space is (up to base loci) to give a linear system on the projective variety.
Yoneda's lemma says that a scheme X determines and is determined by its points.
Let X be a scheme. Its functor of points is the functor
Hom(−,X) : (Affine schemes)op ⟶ Sets
sending an affine scheme Y to the set of scheme maps .
Source officielle
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