Gluing schemes and fiber products

To glue schemes X_i, specify an open subscheme U_ij in X_i for each pair and an isomorphism U_ij→U_ji. The identity and inverse conditions control repeated charts, while a cocycle condition makes the two possible identifications agree on each triple overlap. The domains on which these compositions are defined must also match. These hypotheses produce a scheme covered by the original charts, unique up to the corresponding compatible isomorphism. Merely pairing some points does not supply this structure.

The projective line comes from two affine lines with coordinates t and u by identifying their nonzero opens through u=1/t. Gluing through u=t instead yields the line with a doubled origin, a different scheme with different global behavior. Both gluings use valid affine overlap isomorphisms. Thus gluing schemes does not automatically produce a separated scheme, and a list of identical affine charts does not determine a unique global object without its transition maps.

Given X→S and Y→S, their fiber product represents pairs of maps from any test scheme T that agree after composition to S. The test scheme may have nilpotents and nontrivial residue fields, so restricting this property to ordinary points throws away information. The two projections are part of the construction. A base change is precisely such a product, and the fiber over a point s is obtained by using Spec of its residue field as the new base.

For affine schemes the coordinate ring is the tensor product over the common base ring. Ring maps out of that tensor product correspond to the required compatible pair of ring maps, proving the universal property. General fiber products are assembled by gluing these affine calculations. The underlying topological space is not generally the topological fiber product: over a matching pair of points, primes of the tensor product of their residue fields can introduce several points or new residue extensions.

For a polynomial map to the affine line, taking the fiber over t=c amounts to imposing t=c in the source ring. A repeated root can leave a nonzero nilpotent in the quotient even when the ordinary fiber contains just one point. A reduced fiber at a different parameter may split into distinct points. Counting the vector-space length of a finite fiber can reveal a conserved multiplicity that an unweighted count of points misses.

Residue extensions create a second source of surprises. A field spectrum has one point, but tensoring that field with itself over a smaller field can split. This reflects distinct embeddings compatible with the base rather than a failure of the universal property. In more complicated families the fiber rings may also change dimension or acquire embedded components. Compute the base ring, the algebra maps, and the tensor product before interpreting the geometric picture or claiming that a family preserves an invariant.