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Definition
Pulling an object over a base back along a morphism to a new base.
The map becomes trivial after changing the base from to : its pullback has coordinate ring , which splits into one factor for each Galois automorphism.
Precisely, for a morphism and a new base , the base change of along is
It is formed using the fiber product. A morphism over similarly pulls back to
On affine schemes, if changes the base and , then
In the Galois example, records two copies of constrained to lie over the same base. It is the scheme-theoretic version of “pairs in one fiber.”
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.
An affine scheme is a locally ringed space isomorphic to
for some commutative ring . Here is the prime spectrum with its Zariski topology, and is its structure sheaf. The ring is recovered from global sections:
For example, if is a field, is a one-point affine scheme whose local ring is . The affine line is another affine scheme, but it contains more than the familiar -valued points: it also has points corresponding to other prime ideals, including a generic point. General schemes are assembled by gluing affine schemes along open subsets.