A scheme equipped with a specified morphism to a fixed base scheme.
Let S be a scheme. A scheme over S, or S-scheme, is a scheme X equipped with a specified morphism of schemes
X⟶S,
called its structure morphism.
Given S-schemes X→S and Y→S, an S-morphismf:X→Y is a morphism for which the composite X→Y→S equals the specified map X→S. Schemes over S, together with their S-morphisms, form the category customarily denoted Sch/S.
For example, a ring homomorphism R→A makes SpecA a scheme over SpecR. Constructions such as base change and the fiber product keep track of these specified maps to the base.
In other words, a scheme is assembled by gluing prime spectra of commutative rings, while simultaneously gluing the algebraic functions carried by their structure sheaves.
The map SpecK→SpecF becomes trivial after changing the base from SpecF to SpecK: its pullback has coordinate ring K⊗FK, which splits into one factor for each Galois automorphism.
Precisely, for a morphism X→S and a new base S′→S, the base change of X along S′→S is
XS′:=X×SS′⟶S′.
It is formed using the fiber product. A morphism f:X→Y over S similarly pulls back to
fS′:X×SS′⟶Y×SS′.
On affine schemes, if R→R′ changes the base and X=SpecA, then
In the Galois example, SpecK×SpecFSpecK records two copies of SpecK constrained to lie over the same base. It is the scheme-theoretic version of “pairs in one fiber.”
Given morphisms of schemesX→S←Y, their fiber product is a scheme X×SY with projections to X and Y whose composites to S agree, and which is universal with that property: for every scheme T,
Hom(T,X×SY)≅Hom(T,X)×Hom(T,S)Hom(T,Y).
On affine schemes the construction is concrete. Ring maps R→A and R→B give
SpecA×SpecRSpecB≅Spec(A⊗RB),
where A⊗RB is the tensor product. This affine calculation is why K⊗FK appears in the Galois torsor identity.