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Definition
The fiber of a scheme morphism after base change to an algebraic closure of a residue field.
Let be a morphism of schemes, let , and let be its residue field. Choose an algebraic closure . The geometric fiber over the geometric point is the fiber product
The ordinary fiber is ; the geometric fiber extends its scalars to an algebraic closure.
A morphism of schemes is a morphism of locally ringed spaces. It consists of a continuous map of underlying spaces and a compatible morphism of structure sheaves into a direct image sheaf:
For every , the induced map on stalks
must be a local ring homomorphism: the inverse image of the target's maximal ideal is the source's maximal ideal.
The basic example comes from a ring homomorphism . It induces a morphism of affine schemes
The reversed direction is fundamental: is contravariant. In particular, a field inclusion gives .
Let be a local ring. The residue field of is
It is a field because is maximal.
The canonical surjection
is sometimes called the residue map.
More generally, for a prime ideal , the residue field at is
where is the localization at . This is the residue field of the local ring .
Let be a field.
Definition (algebraic closure). An algebraic closure of is a field equipped with an inclusion such that:
In particular, every splits in , so contains a splitting field for every polynomial over .
Given morphisms of schemes , their fiber product is a scheme with projections to and whose composites to agree, and which is universal with that property: for every scheme ,
On affine schemes the construction is concrete. Ring maps and give
where is the tensor product. This affine calculation is why appears in the Galois torsor identity.