In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.

A morphism of algebraic stacks generalizes a morphism of schemes.

Definition

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By definition, a morphism of schemes is just a morphism of locally ringed spaces. Isomorphisms are defined accordingly.

A scheme, by definition, has open affine charts and thus a morphism of schemes can also be described in terms of such charts (compare the definition of morphism of varieties).[1] Let ƒ:XY be a morphism of schemes. If x is a point of X, since ƒ is continuous, there are open affine subsets U = Spec A of X containing x and V = Spec B of Y such that ƒ(U) ⊆ V. Then ƒ: UV is a morphism of affine schemes and thus is induced by some ring homomorphism BA (cf. #Affine case.) In fact, one can use this description to "define" a morphism of schemes; one says that ƒ:XY is a morphism of schemes if it is locally induced by ring homomorphisms between coordinate rings of affine charts.

  • Note: It would not be desirable to define a morphism of schemes as a morphism of ringed spaces. One trivial reason is that there is an example of a ringed-space morphism between affine schemes that is not induced by a ring homomorphism (for example,[2] a morphism of ringed spaces:
     
that sends the unique point to s and that comes with  .) More conceptually, the definition of a morphism of schemes needs to capture "Zariski-local nature" or localization of rings;[3] this point of view (i.e., a local-ringed space) is essential for a generalization (topos).

Let f : XY be a morphism of schemes with  . Then, for each point x of X, the homomorphism on the stalks:

 

is a local ring homomorphism: i.e.,   and so induces an injective homomorphism of residue fields

 .

(In fact, φ maps th n-th power of a maximal ideal to the n-th power of the maximal ideal and thus induces the map between the (Zariski) cotangent spaces.)

For each scheme X, there is a natural morphism

 

which is an isomorphism if and only if X is affine; θ is obtained by gluing U → target which come from restrictions to open affine subsets U of X. This fact can also be stated as follows: for any scheme X and a ring A, there is a natural bijection:

 

(Proof: The map   from the right to the left is the required bijection. In short, θ is an adjunction.)

Moreover, this fact (adjoint relation) can be used to characterize an affine scheme: a scheme X is affine if and only if for each scheme S, the natural map

 

is bijective.[4] (Proof: if the maps are bijective, then   and X is isomorphic to   by Yoneda's lemma; the converse is clear.)

A morphism as a relative scheme

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Fix a scheme S, called a base scheme. Then a morphism   is called a scheme over S or an S-scheme; the idea of the terminology is that it is a scheme X together with a map to the base scheme S. For example, a vector bundle ES over a scheme S is an S-scheme.

An S-morphism from p:XS to q:YS is a morphism ƒ:XY of schemes such that p = q ∘ ƒ. Given an S-scheme  , viewing S as an S-scheme over itself via the identity map, an S-morphism   is called a S-section or just a section.

All the S-schemes form a category: an object in the category is an S-scheme and a morphism in the category an S-morphism. (This category is the slice category of the category of schemes with the base object S.)

Affine case

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Let   be a ring homomorphism and let

 

be the induced map. Then

  •   is continuous.[5]
  • If   is surjective, then   is a homeomorphism onto its image.[6]
  • For every ideal I of A,  [7]
  •   has dense image if and only if the kernel of   consists of nilpotent elements. (Proof: the preceding formula with I = 0.) In particular, when B is reduced,   has dense image if and only if   is injective.

Let f: Spec A → Spec B be a morphism of schemes between affine schemes with the pullback map  : BA. That it is a morphism of locally ringed spaces translates to the following statement: if   is a point of Spec A,

 .

(Proof: In general,   consists of g in A that has zero image in the residue field k(x); that is, it has the image in the maximal ideal  . Thus, working in the local rings,  . If  , then   is a unit element and so   is a unit element.)

Hence, each ring homomorphism BA defines a morphism of schemes Spec A → Spec B and, conversely, all morphisms between them arise this fashion.

Examples

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Basic ones

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  • Let R be a field or   For each R-algebra A, to specify an element of A, say f in A, is to give a R-algebra homomorphism   such that  . Thus,  . If X is a scheme over S = Spec R, then taking   and using the fact Spec is a right adjoint to the global section functor, we get   where  . Note the equality is that of rings.
  • Similarly, for any S-scheme X, there is the identification of the multiplicative groups:   where   is the multiplicative group scheme.
  • Many examples of morphisms come from families parameterized by some base space. For example,   is a projective morphism of projective varieties where the base space parameterizes quadrics in  .

Graph morphism

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Given a morphism of schemes   over a scheme S, the morphism   induced by the identity   and f is called the graph morphism of f. The graph morphism of the identity is called the diagonal morphism.

Types of morphisms

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Finite type

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Morphisms of finite type are one of the basic tools for constructing families of varieties. A morphism   is of finite type if there exists a cover   such that the fibers   can be covered by finitely many affine schemes   making the induced ring morphisms   into finite-type morphisms. A typical example of a finite-type morphism is a family of schemes. For example,

 

is a morphism of finite type. A simple non-example of a morphism of finite-type is   where   is a field. Another is an infinite disjoint union

 

Closed immersion

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A morphism of schemes   is a closed immersion if the following conditions hold:

  1.   defines a homeomorphism of   onto its image
  2.   is surjective

This condition is equivalent to the following: given an affine open   there exists an ideal   such that  

Examples

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Of course, any (graded) quotient   defines a subscheme of   ( ). Consider the quasi-affine scheme   and the subset of the  -axis contained in  . Then if we take the open subset   the ideal sheaf is   while on the affine open   there is no ideal since the subset does not intersect this chart.

Separated

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Separated morphisms define families of schemes which are "Hausdorff". For example, given a separated morphism   in   the associated analytic spaces   are both Hausdorff. We say a morphism of scheme   is separated if the diagonal morphism   is a closed immersion. In topology, an analogous condition for a space   to be Hausdorff is if the diagonal set

 

is a closed subset of  . Nevertheless, most schemes are not Hausdorff as topological spaces, as the Zariski topology is in general highly non-Hausdorff.

Examples

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Most morphisms encountered in scheme theory will be separated. For example, consider the affine scheme

 

over   Since the product scheme is

 

the ideal defining the diagonal is generated by

 

showing the diagonal scheme is affine and closed. This same computation can be used to show that projective schemes are separated as well.

Non-examples

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The only time care must be taken is when you are gluing together a family of schemes. For example, if we take the diagram of inclusions

 

then we get the scheme-theoretic analogue of the classical line with two-origins.

Proper

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A morphism   is called proper if

  1. it is separated
  2. of finite-type
  3. universally closed

The last condition means that given a morphism   the base change morphism   is a closed immersion. Most known examples of proper morphisms are in fact projective; but, examples of proper varieties which are not projective can be found using toric geometry.

Projective

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Projective morphisms define families of projective varieties over a fixed base scheme. Note that there are two definitions: Hartshornes which states that a morphism   is called projective if there exists a closed immersion   and the EGA definition which states that a scheme   is projective if there is a quasi-coherent  -module of finite type such that there is a closed immersion  . The second definition is useful because an exact sequence of   modules can be used to define projective morphisms.

Projective morphism over a point

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A projective morphism   defines a projective scheme. For example,

 

defines a projective curve of genus   over  .

Family of projective hypersurfaces

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If we let   then the projective morphism

 

defines a family of Calabi-Yau manifolds which degenerate.

Lefschetz pencil

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Another useful class of examples of projective morphisms are Lefschetz Pencils: they are projective morphisms   over some field  . For example, given smooth hypersurfaces   defined by the homogeneous polynomials   there is a projective morphism

 

giving the pencil.

EGA projective

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A nice classical example of a projective scheme is by constructing projective morphisms which factor through rational scrolls. For example, take   and the vector bundle  . This can be used to construct a  -bundle   over  . If we want to construct a projective morphism using this sheaf we can take an exact sequence, such as

 

which defines the structure sheaf of the projective scheme   in  

Flat

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Intuition

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Flat morphisms have an algebraic definition but have a very concrete geometric interpretation: flat families correspond to families of varieties which vary "continuously". For example,

 

is a family of smooth affine quadric curves which degenerate to the normal crossing divisor

 

at the origin.

Properties

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One important property that a flat morphism must satisfy is that the dimensions of the fibers should be the same. A simple non-example of a flat morphism then is a blowup since the fibers are either points or copies of some  .

Definition

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Let   be a morphism of schemes. We say that   is flat at a point   if the induced morphism   yields an exact functor   Then,   is flat if it is flat at every point of  . It is also faithfully flat if it is a surjective morphism.

Non-example

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Using our geometric intuition it obvious that

 

is not flat since the fiber over   is   with the rest of the fibers are just a point. But, we can also check this using the definition with local algebra: Consider the ideal   Since   we get a local algebra morphism

 

If we tensor

 

with  , the map

 

has a non-zero kernel due the vanishing of  . This shows that the morphism is not flat.

Unramified

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A morphism   of affine schemes is unramified if  . We can use this for the general case of a morphism of schemes  . We say that   is unramified at   if there is an affine open neighborhood   and an affine open   such that   and   Then, the morphism is unramified if it is unramified at every point in  .

Geometric example

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One example of a morphism which is flat and generically unramified, except for at a point, is

 

We can compute the relative differentials using the sequence

 

showing

 

if we take the fiber  , then the morphism is ramified since

 

otherwise we have

 

showing that it is unramified everywhere else.

Etale

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A morphism of schemes   is called étale if it is flat and unramfied. These are the algebro-geometric analogue of covering spaces. The two main examples to think of are covering spaces and finite separable field extensions. Examples in the first case can be constructed by looking at branched coverings and restricting to the unramified locus.

Morphisms as points

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By definition, if X, S are schemes (over some base scheme or ring B), then a morphism from S to X (over B) is an S-point of X and one writes:

 

for the set of all S-points of X. This notion generalizes the notion of solutions to a system of polynomial equations in classical algebraic geometry. Indeed, let X = Spec(A) with  . For a B-algebra R, to give an R-point of X is to give an algebra homomorphism AR, which in turn amounts to giving a homomorphism

 

that kills fi's. Thus, there is a natural identification:

 

Example: If X is an S-scheme with structure map π: XS, then an S-point of X (over S) is the same thing as a section of π.

In category theory, Yoneda's lemma says that, given a category C, the contravariant functor

 

is fully faithful (where   means the category of presheaves on C). Applying the lemma to C = the category of schemes over B, this says that a scheme over B is determined by its various points.

It turns out that in fact it is enough to consider S-points with only affine schemes S, precisely because schemes and morphisms between them are obtained by gluing affine schemes and morphisms between them. Because of this, one usually writes X(R) = X(Spec R) and view X as a functor from the category of commutative B-algebras to Sets.

Example: Given S-schemes X, Y with structure maps p, q,

 .

Example: With B still denoting a ring or scheme, for each B-scheme X, there is a natural bijection

  { the isomorphism classes of line bundles L on X together with n + 1 global sections generating L. };

in fact, the sections si of L define a morphism  . (See also Proj construction#Global Proj.)

Remark: The above point of view (which goes under the name functor of points and is due to Grothendieck) has had a significant impact on the foundations of algebraic geometry. For example, working with a category-valued (pseudo-)functor instead of a set-valued functor leads to the notion of a stack, which allows one to keep track of morphisms between points (i.e., morphisms between morphisms).

Rational map

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A rational map of schemes is defined in the same way for varieties. Thus, a rational map from a reduced scheme X to a separated scheme Y is an equivalence class of a pair   consisting of an open dense subset U of X and a morphism  . If X is irreducible, a rational function on X is, by definition, a rational map from X to the affine line   or the projective line  

A rational map is dominant if and only if it sends the generic point to the generic point.[8]

A ring homomorphism between function fields need not induce a dominant rational map (even just a rational map).[9] For example, Spec k[x] and Spec k(x) and have the same function field (namely, k(x)) but there is no rational map from the former to the latter. However, it is true that any inclusion of function fields of algebraic varieties induces a dominant rational map (see morphism of algebraic varieties#Properties.)

See also

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Notes

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  1. ^ Vakil 2014, Exercise 6.3.C.
  2. ^ Vakil 2014, Exercise 6.2.E.
  3. ^ Derived Algebraic Geometry V: Structured Spaces (PDF), 22 February 2011, § 1.
  4. ^ Grothendieck & Dieudonné 1960, Ch. I, Corollarie 1.6.4.
  5. ^ Proof:   for all f in A.
  6. ^ Grothendieck & Dieudonné 1960, Ch. I, Corollaire 1.2.4.
  7. ^ Grothendieck & Dieudonné 1960, Ch. I, 1.2.2.3.
  8. ^ Vakil 2014, Exercise 6.5.A
  9. ^ Vakil 2014, A paragraph after Exercise 6.5.B

References

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