In algebraic geometry, a functor represented by a schemeX is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme S is 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, 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. Some moduli problems are solved by giving formal 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 spaceS is 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 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 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 : op ⟶ Sets sending an affine schemeA 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:Schemesop → Sets. Conversely, a functor F:op → Sets is the functor of points of some scheme if and only ifF is a sheaf with respect to the Zariski topology on, 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 . For example, if X is an affine scheme Spec and x is a prime ideal, then the residue field of x is the function field of the closed subscheme. For simplicity, suppose. Then the inclusion of a set-theoretic point x into X corresponds to the ring homomorphism:
Points as sections
By the universal property offiber product, each R-point of a scheme X determines a morphism of R-schemes i.e., a section of the projection. If S is a subset of X, then one writes for the set of the images of the sections determined by elements in S.
Let, the Spec of the ring of dual numbers over a field k and X a scheme over k. Then each amounts to the tangent vector to X at the point that is the image of the closed point of the map. In other words, is the set of tangent vectors to X.
