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Concept
Morphism of schemes
Summary
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.
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 f : X → Y 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 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.
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