Maps between schemes, base change, and families of geometric objects
A morphism of schemes is a map of spaces that also pulls back functions. Its fibers can be fatter than plain preimages: they remember where points collide.
Drag the base point y, or press Animate, to move along the base of the parabola y = x². Watch the fiber go from two points, to a single fat point at y = 0, to empty.
A morphism of schemes f: X → Y is a continuous map of topological spaces together with a compatible map of structure sheaves f#: 𝒪Y → f*𝒪X. The sheaf map ensures that "pulling back functions" makes algebraic sense — it is the geometric counterpart of a ring homomorphism. For affine schemes, a morphism Spec(S) → Spec(R) is exactly a ring map R → S, with the geometric direction reversed.
A morphism f: X → Y consists of a continuous map of the underlying topological spaces and a sheaf map f# that pulls regular functions on Y back to regular functions on X. The crucial requirement is that f#preserves the local ring structure: for each point x ∈ X, the induced map on stalks 𝒪Y,f(x) → 𝒪X,x sends the maximal ideal of 𝒪Y,f(x) into the maximal ideal of 𝒪X,x. This "locally ringed" condition ensures that morphisms respect vanishing of functions.
Important classes of morphisms include open immersions(inclusions of open subschemes), closed immersions (surjections on structure sheaves), and finite morphisms (where the pushforward of 𝒪X is a finite 𝒪Y-module). Each class captures a different geometric relationship between source and target.
The fiber product X ×S Y is the scheme-theoretic analog of the pullback or intersection. It is the universal scheme mapping to both X and Y compatibly over S. Fibers of a morphism f: X → Y at a point y are scheme-theoretic pullbacks X ×Y Spec(k(y)), and they can be "fatter" than set-theoretic preimages — scheme theory sees tangency and multiplicity that the set-theoretic fiber misses.
The fiber product X ×S Y is defined by a universal property: it is the terminal object among schemes mapping to both X and Y over S. In the affine case, Spec(A) ×Spec(R) Spec(B) = Spec(A ⊗R B) — the geometric fiber product is just the tensor product of rings. This clean algebraic description is one of the great advantages of scheme theory.
Base change is the operation of pulling back a morphism f: X → S along a map T → S, producing X ×S T → T. This lets you "re-parameterize" a family of geometric objects: for example, base-changing a variety over ℚ to ℚp gives the p-adic fiber. Properties like flatness, smoothness, and properness are designed to behave well under base change.
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