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.
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More generally, morphisms p:X →S with various schemes X but fixed scheme S form the category of schemes over S (the slice category of the category of schemes with the base object S.) An object in the category is called an S-scheme and a morphism in the category an S-morphism; explicitly, an S-morphism from p:X →S to q:Y →S is a morphism ƒ:X →Y of schemes such that p = q ∘ ƒ.
Definition
By definition, a morphism of schemes is just a morphism of locally ringed spaces.
A scheme, by definition, has an open affine chart 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.
Let ƒ:X→Y be a morphism of schemes with
is a local ring homomorphism: i.e.,
(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
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
Affine case
Let
Let ƒ: Spec A → Spec B be a morphism of schemes between affine schemes with the pullback map φ: B → A. That it is a morphism of locally ringed spaces translates to the following statement: if
(Proof: In general,
Hence, each ring homomorphism B → A defines a morphism of schemes Spec A → Spec B and, conversely, all morphisms between them arise this fashion.
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. This notion generalizes the notion of solutions to a system of polynomial equations in classical algebraic geometry. Indeed, let X = Spec(A) with
that kills fi'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 the category theory, Yoneda's lemma says that, given a category C, the contravariant functor
is fully faithful (where
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 si of L define a morphism
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.
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
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.)
Types of morphisms
For now, see glossary of algebraic geometry.