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Concept
Base change theorems
Summary
In mathematics, the base change theorems relate the and the of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves:
where
is a of topological spaces and is a sheaf on X.
Such theorems exist in different branches of geometry: for (essentially arbitrary) topological spaces and proper maps f, in algebraic geometry for (quasi-)coherent sheaves and f proper or g flat, similarly in analytic geometry, but also for étale sheaves for f proper or g smooth.
A simple base change phenomenon arises in commutative algebra when A is a commutative ring and B and A' are two A-algebras. Let . In this situation, given a B-module M, there is an isomorphism (of A' -modules):
Here the subscript indicates the forgetful functor, i.e., is M, but regarded as an A-module.
Indeed, such an isomorphism is obtained by observing
Thus, the two operations, namely forgetful functors and tensor products commute in the sense of the above isomorphism.
The base change theorems discussed below are statements of a similar kind.
The base change theorems presented below all assert that (for different types of sheaves, and under various assumptions on the maps involved), that the following base change map
is an isomorphism, where
are continuous maps between topological spaces that form a and is a sheaf on X. Here denotes the of under f, i.e., the derived functor of the direct image (also known as pushforward) functor .
This map exists without any assumptions on the maps f and g. It is constructed as follows: since is left adjoint to , there is a natural map (called unit map)
and so
The Grothendieck spectral sequence then gives the first map and the last map (they are edge maps) in:
Combining this with the above yields
Using the adjointness of and finally yields the desired map.
The above-mentioned introductory example is a special case of this, namely for the affine schemes and, consequently, , and the quasi-coherent sheaf associated to the B-module M.
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