Functor represented by a scheme

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a functor represented by a scheme X is a set-valued <a href="/facts/Contravariant_functor/l2u3rIBE">contravariant functor</a> on the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of <a href="/facts/Scheme_(mathematics)/c7uSqGqL">schemes</a> such that the value of the functor at each scheme S is (up to natural <a href="/facts/Bijection/j4gTuTmW">bijections</a>, or one-to-one correspondence) the set of all <a href="/facts/Morphism_of_schemes/55CKqHHo">morphisms</a> 
 
 
 
 S
 →
 X
 
 
 {\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 <a href="/facts/Representable_functor/dILwN4oj">represent</a> the functor F, and to classify geometric objects over S given by F.
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 <a href="/facts/Line_bundle/CupAl1ft">line bundles</a> over S (or more precisely n-dimensional <a href="/facts/Linear_system_of_divisors/L8zleDlj">linear systems</a>) is represented by the <a href="/facts/Projective_space/7MSpA9cM">projective space</a> 
 
 
 
 X
 =
 
 
 P
 
 
 n
 −
 1
 
 
 
 
 {\displaystyle X=\mathbb {P} ^{n-1}}
 
. Another example is the <a href="/facts/Hilbert_scheme/kJcbts4e">Hilbert scheme</a> X of a scheme Y, which represents the functor sending a scheme S to the set of closed subschemes of 
 
 
 
 Y
 ×
 S
 
 
 {\displaystyle Y\times S}
 
 which are <a href="/facts/Flat_morphism/CLM7e8d6">flat families</a> over S. 
In some applications, it may not be possible to find a scheme that represents a given functor. This led to the notion of a <a href="/facts/Stack_(mathematics)/MkASN1rA">stack</a>, which is <a href="/facts/Pseudo-functor/15bRbrVn">not quite a functor</a> 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, <a href="/facts/Deformation_theory/tVLS3kkG">deformation theory</a> 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 <a href="/facts/Formal_scheme/FCB3BMez">formal scheme</a>. 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.

Functor represented by a scheme open-in-new

Functor represented by a scheme