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 ${\displaystyle S\to X}$. The functor F is then said to be naturally equivalent to the functor of points of X; and the scheme X is said to represent the functor F, and to classify geometric objects over S given by F.^[1]

A functor producing certain geometric objects over S might be represented by a scheme X. For example, the functor taking S to the set of all line bundles over S (or more precisely n-dimensional linear systems) is represented by the projective space ${\displaystyle X=\mathbb {P} ^{n-1}}$. Another example is the Hilbert scheme X of a scheme Y, which represents the functor sending a scheme S to the set of closed subschemes of ${\displaystyle Y\times S}$ which are flat families over S.^[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 rather than 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 a scheme that can represent the same functor, up to some isomorphisms.