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Unformatted text preview: FIBER PRODUCTS; SEPARATED AND PROPER MORPHISMS PETE L. CLARK 1. Fiber products and base change A very important tool across geometry is the fibered product. Here the basic state- ment is simple: given two S-schemes X and Y , the fiber product X S Y exists in the category of schemes. Namely, this is an S-scheme, endowed with S-morphisms X , Y to X and Y , satisfying the following universal mapping property: given any S-scheme Z and S-morphisms f : Z X , g : Z Y , there exists a unique morphism f S g : Z X S Y making the usual diagram (see e.g. Liu, p. 79) commutative. As usual, the first thing to do is consider the affine case with all the arrows re- versed. In this case, the object which makes the diagram commute is nothing other than the tensor product R A S . The general case is, as usual, reduced to the affine case by glueing (and, as is often the case, this requires some nontrivial work). Most often we take fiber products over an affine scheme Spec A , and we will abuse notation by writing X A Y rather than X Spec A Y . In particular, for any field (or indeed any ring) k we have A m k k A n k = A m + n k , this being the geometric analogue of the usual statement about tensor products of polynomial rings. A useful fact, which follows immediately from the universal property, is: ( X S Y )( S ) = X ( S ) Y ( S ) . If X is an S-scheme and S is another S-scheme, then we can form the fiber prod- uct X S S . This is still an S-scheme, but it is also an S-scheme via the second projection map. This process preciesly, of starting with a scheme over S taking the fiber product with a second scheme S over S , and regarding the fiber product as an S-scheme, is called base change . We denote the base change by X /S . In the case of a field extension k , l and a k-scheme X , the base change X /l is a scheme over l . When X is an affine k-variety, this recovers our previous process of base extension by tensorization from k to l . Note that in general a base change from a Spec R scheme to a Spec S involves a homomorphism R S which need not be injective, hence we use the term base change rather than base extension. An extremely important application of base change is to the notion of fibers of a morphism. Let f : X Y be a morphism of schemes, and y Y be any point (closed or otherwise). We want to define a scheme X y , the fiber of f over y in a 1 2 PETE L. CLARK way which generalizes the usual idea of fibers of a map of sets. The key to this is to use the canonical morphism Spec k ( y ) , Y . Then we can define X y := X Y Spec k ( y ) , which is a scheme over the field Spec k ( y )....
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