Base change

change of base

A category-theoretical construction; special cases are the concept of an induced fibration in topology, and extension of the ring of scalars in the theory of modules.

Let be a category with fibred products and let be a morphism of . A base change by means of is a functor from the category of -objects (i.e. the category of morphism , where is an object of ) to the category of -objects, taking an -object to the -object , where and is projection onto the second factor. The morphism is then called the base-change morphism. One also says that is obtained from by base change.

A special case of a base change is the concept of a fibre of a morphism of schemes : The fibre of the morphism over a point is the scheme

i.e. the scheme obtained from by base change via the natural morphism . A similar definition yields the geometric fibre ; it is obtained by base change via the morphism associated with a geometric point of , where is an algebraically closed field. Many properties of the -scheme are preserved under a base change. The inverse problem — to infer the properties of a scheme from those of the schemes obtained from by base change — is considered in descent theory (see also [3]).

Let be the morphism obtained from via a morphism , so that one has a Cartesian square

Let be a sheaf of sets on . Then there exists a natural sheaf mapping . If is a sheaf of Abelian groups, then for every there exists a natural sheaf homomorphism

Under these conditions, and are also called base-change morphisms. It is usually said that the base-change theorem is valid if (or ) is an isomorphism. In other words, the base-change theorem is a proposition about the compatibility (commutability) of the functors with the base-change functor. In particular, if is the imbedding of a point , the base-change theorem states that there exists a natural isomorphism between the fibre of the -th direct image of the sheaf and the -dimensional cohomology group of the fibre of the morphism . The base-change theorem is valid in the following situations: 1) is a proper mapping of paracompact topological spaces, is a locally compact space [1]; 2) is a separable quasi-compact morphism of schemes, is a flat morphism, is a quasi-coherent sheaf of -modules (the comparison theorem for the cohomology of ordinary and formal schemes — see [2] — can also be interpreted as a base-change theorem); or 3) is a proper morphism of schemes, is a torsion sheaf in the étale topology. Some other cases in which base-change theorems are valid are considered in [3].

References

[1]	R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958)
[2]	A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique" Publ. Math. IHES , 11 (1961)
[3]	"Théorie des topos et cohomologie étale des schémas" M. Artin (ed.) A. Grothendieck (ed.) J.-L. Verdier (ed.) , Sem. Geom. Alg. , 4 , Springer (1973)