Functor represented by a scheme

In algebraic geometry, a functor represented by a scheme X is a set-valued contravarinat 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 $S\to X$. The scheme X is then said to represent the functor and that classify geometric objects over S.

The best known example is that of 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 $X\times S$.

In some applications, it may not be possible to find a scheme that represents a given functor. This led to the notion of a stack, which is a sort like a functor but can still be treated like as if it were a geometric object. (A Hilbert scheme is a scheme not a stack because, very roughly speaking, deformation theory is simpler for closed schemes.)

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 $EG\to BG$ along some map from S to $BG$. 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 $BG$ 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.

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 ${\mathcal {O}}_{X,x}$ (i.e., the quotient by the maximal ideal). For example, if X is an affine scheme Spec(A) and x is a prime ideal ${\mathfrak {p}}$, then the residue field of x is the function field of the closed subscheme $\operatorname {Spec} (A/{\mathfrak {p}})$.

For simplicity, suppose $X=\operatorname {Spec} (A)$. Then the inclusion of a set-theoretic point x into X corresponds to the ring homomorphism:

$A\to k(x)$

(which is $A\to A_{\mathfrak {p}}\to k({\mathfrak {p}})$ if $x={\mathfrak {p}}$.)

Points as sections

By the universal property of fiber product, each R-point of a scheme X determines a morphism of R-schemes

$\operatorname {Spec} (R)\to X_{R}{\overset {\mathrm {def} }{=}}X\times _{\operatorname {Spec} (B)}\operatorname {Spec} (R)$;

i.e., a section of the projection $X_{R}\to \operatorname {Spec} (R)$. If S is a subset of X(R), then one writes $|S|\subset X_{R}$ for the set of the images of the sections determined by elements in S.

Spec of the ring of dual numbers

Let $D=\operatorname {Spec} (k[t]/(t^{2}))$, the Spec of the ring of dual numbers over a field k and X a scheme over k. Then each $D\to X$ amounts to the tangent vector to X at the point that is the image of the closed point of the map. In other words, $X(D)$ is the set of tangent vectors to X.

Universal object

Let F be the functor represented by a scheme X. Under the isomorphism $F(X)\simeq \operatorname {Mor} (X,X)$, there is a unique element of $F(X)$ that corresponds to the identity map $1_{X}:X\to X$. It is called the universal object or the universal family (when the objects that are being classified are families).

See also

• Weil restriction
• Rational point
• descent along torsors.

References

• David Mumford (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians (2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 3-540-63293-X.
• http://www.math.harvard.edu/~lurie/282ynotes/LectureXIV-Borel.pdf
• Shafarevich, Igor (1994). Basic Algebraic Geometry, Second, revised and expanded edition, Vol. 2. Springer-Verlag.

External links

• http://www.math.washington.edu/~zhang/Shanghai2011/Slides/ardakov.pdf