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).
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.
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
, viewing S as an S-scheme over itself via the identity map, an S-morphism
That it is a morphism of locally ringed spaces translates to the following statement: if
Morphisms of finite type are one of the basic tools for constructing families of varieties.
A typical example of a finite-type morphism is a family of schemes.
Another is an infinite disjoint union A morphism of schemes
Separated morphisms define families of schemes which are "Hausdorff".
Most morphisms encountered in scheme theory will be separated.
This same computation can be used to show that projective schemes are separated as well.
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.
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.
Note that there are two definitions: Hartshornes which states that a morphism
For example, defines a projective curve of genus
then the projective morphism defines a family of Calabi-Yau manifolds which degenerate.
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.
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
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
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.
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
A rational map of schemes is defined in the same way for varieties.
However, it is true that any inclusion of function fields of algebraic varieties induces a dominant rational map (see morphism of algebraic varieties#Properties.)