In algebraic geometry, a morphism of schemes generalizes a
morphism of algebraic varieties In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regul ...
just as a
scheme generalizes an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. It is, by definition, a morphism in the category of schemes.
A
morphism of algebraic stacks In algebraic geometry, given algebraic stacks p: X \to C, \, q: Y \to C over a base category ''C'', a morphism f: X \to Y of algebraic stacks is a functor such that q \circ f = p.
More generally, one can also consider a morphism between prestacks; ...
generalizes a morphism of schemes.
Definition
By definition, a morphism of schemes is just a morphism of
locally ringed spaces.
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). Let ƒ:''X''→''Y'' 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: ''U'' → ''V'' is a morphism of
affine schemes and thus is induced by some ring homomorphism ''B'' → ''A'' (cf.
#Affine case.) In fact, one can use this description to "define" a morphism of schemes; one says that ƒ:''X''→''Y'' 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, 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; this point of view (i.e., a local-ringed space) is essential for a generalization (topos).
Let be a morphism of schemes with
. Then, for each point ''x'' of ''X'', the homomorphisms on the stalks:
:
is a
local ring homomorphism: i.e.,
and so induces an injective homomorphism of
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
s
:
.
(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. (Proof: if the maps are bijective, then
and ''X'' is isomorphic to
by
Yoneda's lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...
; the converse is clear.)
A morphism as a relative scheme
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 ''E'' → ''S'' over a scheme ''S'' is an ''S''-scheme.
An ''S''-morphism from ''p'':''X'' →''S'' to ''q'':''Y'' →''S'' is a morphism ƒ:''X'' →''Y'' 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. (Succinctly, this category is the
slice category
In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track ...
of the category of schemes with the base object ''S''.)
Affine case
Let
be a ring homomorphism and let
:
be the induced map. Then
*
is continuous.
*If
is surjective, then
is a homeomorphism onto its image.
*For every ideal ''I'' of ''A'',
*
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
: ''B'' → ''A''. 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 In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
''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 ''B'' → ''A'' defines a morphism of schemes Spec ''A'' → Spec ''B'' and, conversely, all morphisms between them arise this fashion.
Examples
Basic ones
*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
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
In category theory, a branch of mathematics, for any object a in any category \mathcal where the product a\times a exists, there exists the diagonal morphism
:\delta_a : a \rightarrow a \times a
satisfying
:\pi_k \circ \delta_a = \operatorn ...
.
Types of morphisms
Finite Type
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
A morphism of schemes
is a
closed immersion
In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formaliz ...
if the following conditions hold:
#
defines a homeomorphism of
onto its image
#
is surjective
This condition is equivalent to the following: given an affine open
there exists an ideal
such that
Examples
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
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
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
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
A morphism
is called
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
if
# it is separated
# of finite-type
# 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 In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be n ...
.
Projective
Projective morphisms define families of
projective varieties
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
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
A projective morphism
defines a projective scheme. For example,
:
defines a projective curve of genus
over
.
Family of Projective Hypersurfaces
If we let
then the projective morphism
:
defines a family of Calabi-Yau manifolds which degenerate.
Lefschetz Pencil
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
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
Intuition
Flat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,
:f_P\colon \mathcal_ ...
s 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
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
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
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
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
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
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
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 ''A'' →''R'', which in turn amounts to giving a homomorphism
:
that kills ''f''
''i'''s. Thus, there is a natural identification:
:
Example: If ''X'' is an ''S''-scheme with structure map π: ''X'' → ''S'', then an ''S''-point of ''X'' (over ''S'') is the same thing as a section of π.
In
category theory,
Yoneda's lemma
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...
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
:
;
in fact, the sections ''s''
''i'' 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 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. T ...
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.
Rational map
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.
A ring homomorphism between function fields need not induce a dominant rational map (even just a rational map).
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
*
Regular embedding
In algebraic geometry, a closed immersion i: X \hookrightarrow Y of schemes is a regular embedding of codimension ''r'' if each point ''x'' in ''X'' has an open affine neighborhood ''U'' in ''Y'' such that the ideal of X \cap U is generated by a re ...
*
Constructible set (topology)
In topology, constructible sets are a class of subsets of a topological space that have a relatively "simple" structure.
They are used particularly in algebraic geometry and related fields. A key result known as ''Chevalley's theorem''
in algebraic ...
*
Universal homeomorphism In algebraic geometry, a universal homeomorphism is a morphism of schemes f: X \to Y such that, for each morphism Y' \to Y, the base change X \times_Y Y' \to Y' is a homeomorphism of topological spaces.
A morphism of schemes is a universal homeomor ...
Notes
References
*
*{{Hartshorne AG
*Milne, Review of Algebraic Geometry a
Algebraic Groups: The theory of group schemes of finite type over a field.*Vakil
Foundations of Algebraic Geometry
Algebraic geometry