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 $S\to X$ . The scheme X is then said to represent the functor and that classify geometric objects over S given by F.^[1]

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 $X\times S$ .^[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, but 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 $EG\to BG$ 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.^[3]

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 Y to the set of scheme maps $Y\to X$ .^[4]

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):Schemes^op → 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.^[5]

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.^[6]

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.^[1] 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).^[1]

Notes

1 2 3 Shafarevich 1994, Ch. VI § 4.1.
↑ Shafarevich 1994, Ch. VI § 4.4.
↑ In fact, X is determined by its R-points with various rings R: in the precise terms, given schemes X, Y, any natural transformation from the functor $R\mapsto X(R)$ to the functor $R\mapsto Y(R)$ determines a morphism of schemes X →Y in a natural way.
↑ The Stacks Project, 01J5
↑ The functor of points, Yoneda's lemmma, moduli spaces and universal properties (Brian Osserman), Cor. 3.6
↑ This seems like a standard notation; see for example "Nonabelian Poincare Duality in Algebraic Geometry (Lecture 9)" (PDF).

References

David Mumford (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians. Lecture Notes in Mathematics. Vol. 1358 (2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 3-540-63293-X.
Lurie, Jacob. "Lecture 14: Existence of Borel Reductions (I)" (PDF).
Shafarevich, Igor (1994). Basic Algebraic Geometry, Second, revised and expanded edition, Vol. 2. Springer-Verlag.
Shafarevich, Igor R. (2013). Basic Algebraic Geometry 2. doi:10.1007/978-3-642-38010-5. ISBN 978-3-642-38009-9.

External links

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

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[Schemes_and_functors-1] 1 2 3 Shafarevich 1994, Ch. VI § 4.1.

[2] Shafarevich 1994, Ch. VI § 4.4.

[3] In fact, X is determined by its R-points with various rings R: in the precise terms, given schemes X, Y, any natural transformation from the functor $R\mapsto X(R)$ to the functor $R\mapsto Y(R)$ determines a morphism of schemes X →Y in a natural way.

[4] The Stacks Project, 01J5

[5] The functor of points, Yoneda's lemmma, moduli spaces and universal properties (Brian Osserman), Cor. 3.6

[6] This seems like a standard notation; see for example "Nonabelian Poincare Duality in Algebraic Geometry (Lecture 9)" (PDF).

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