Functor represented by a scheme

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 formula_1. 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 formula_2.
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 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.
Motivation.
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 formula_3 along some map from "S" to formula_4. 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 formula_4 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.
Functor of points.
Let "X" be a scheme. Its functor of points is the functor
Hom(−,"X") : (Affine schemes)op ⟶ Sets
sending an affine scheme "A" to the set of scheme maps "A" → "X".
A scheme is determined up to isomorphism by its functor of points. This is a stronger version of the Yoneda lemma, which says that a "X" is determined by the map Hom(−,"X"):Schemesop → Sets.
Conversely, a functor "F":(Affine schemes)op → Sets is the functor of points of some scheme if and only if "F" is a sheaf with respect to the Zariski topology on (Affine schemes), and "F" admits an open cover by affine schemes.
Examples.
Points as characters.
Let "X" be a scheme over the base ring "B". If "x" is a set-theoretic point of "X", then the residue field of "x" is the residue field of the local ring formula_6 (i.e., the quotient by the maximal ideal). For example, if "X" is an affine scheme Spec("A") and "x" is a prime ideal formula_7, then the residue field of "x" is the function field of the closed subscheme formula_8.
For simplicity, suppose formula_9. Then the inclusion of a set-theoretic point "x" into "X" corresponds to the ring homomorphism:
Points as sections.
By the universal property of fiber product, each "R"-point of a scheme "X" determines a morphism of "R"-schemes
i.e., a section of the projection formula_14. If "S" is a subset of "X"("R"), then one writes formula_15 for the set of the images of the sections determined by elements in "S".
Spec of the ring of dual numbers.
Let formula_16, the Spec of the ring of dual numbers over a field "k" and "X" a scheme over "k". Then each formula_17 amounts to the tangent vector to "X" at the point that is the image of the closed point of the map. In other words, formula_18 is the set of tangent vectors to "X".
Universal object.
Let "F" be the functor represented by a scheme "X". Under the isomorphism formula_19, there is a unique element of formula_20 that corresponds to the identity map formula_21. It is called the universal object or the universal family (when the objects that are being classified are families).