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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b0153101.png" /> be a category with fibred products and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b0153102.png" /> be a morphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b0153103.png" />. A base change by means of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b0153104.png" /> is a functor from the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b0153105.png" />-objects (i.e. the category of morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b0153106.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b0153107.png" /> is an object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b0153108.png" />) to the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b0153109.png" />-objects, taking an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531010.png" />-object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531011.png" /> to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531012.png" />-object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531015.png" /> is projection onto the second factor. The morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531016.png" /> is then called the base-change morphism. One also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531017.png" /> is obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531018.png" /> by base change.

A special case of a base change is the concept of a fibre of a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531019.png" /> of schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531020.png" />: The fibre of the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531021.png" /> over a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531022.png" /> is the scheme

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531023.png" />

i.e. the scheme obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531024.png" /> by base change via the natural morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531025.png" />. A similar definition yields the geometric fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531026.png" />; it is obtained by base change via the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531027.png" /> associated with a geometric point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531029.png" /> is an algebraically closed field. Many properties of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531030.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531031.png" /> are preserved under a base change. The inverse problem — to infer the properties of a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531032.png" /> from those of the schemes obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531033.png" /> by base change — is considered in descent theory (see also [3]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531034.png" /> be the morphism obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531035.png" /> via a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531036.png" />, so that one has a Cartesian square

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531037.png" />

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531038.png" /> be a sheaf of sets on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531039.png" />. Then there exists a natural sheaf mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531040.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531041.png" /> is a sheaf of Abelian groups, then for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531042.png" /> there exists a natural sheaf homomorphism

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531043.png" />

Under these conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531045.png" /> are also called base-change morphisms. It is usually said that the base-change theorem is valid if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531046.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531047.png" />) is an isomorphism. In other words, the base-change theorem is a proposition about the compatibility (commutability) of the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531048.png" /> with the base-change functor. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531049.png" /> is the imbedding of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531050.png" />, the base-change theorem states that there exists a natural isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531051.png" /> between the fibre of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531052.png" />-th direct image of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531053.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531054.png" />-dimensional cohomology group of the fibre of the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531055.png" />. The base-change theorem is valid in the following situations: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531056.png" /> is a proper mapping of paracompact topological spaces, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531057.png" /> is a locally compact space [1]; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531058.png" /> is a separable quasi-compact morphism of schemes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531059.png" /> is a flat morphism, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531060.png" /> is a quasi-coherent sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531061.png" />-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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531062.png" /> is a proper morphism of schemes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015310/b01531063.png" /> 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 géométrie 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)