More precisely, they are about the base change map, given by the following natural transformation of sheaves: where is a Cartesian square of topological spaces and
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.
In this situation, given a B-module M, there is an isomorphism (of A' -modules): Here the subscript indicates the forgetful functor, i.e.,
The base change theorems discussed below are statements of a similar kind.
The above-mentioned introductory example is a special case of this, namely for the affine schemes
associated to the B-module M. It is conceptually convenient to organize the above base change maps, which only involve only a single higher direct image functor, into one which encodes all
In fact, similar arguments as above yield a map in the derived category of sheaves on S': where
into one entity, the above statement may equivalently be rephrased by saying that the base change map is a quasi-isomorphism.
The assumptions that the involved spaces be Hausdorff have been weakened by Schnürer & Soergel (2016).
Lurie (2009) has extended the above theorem to non-abelian sheaf cohomology, i.e., sheaves taking values in simplicial sets (as opposed to abelian groups).
(or its derived functor) has to be replaced by the direct image with compact support
The proper base change theorem mentioned above has the following generalization: there is a quasi-isomorphism[4] Proper base change theorems for quasi-coherent sheaves apply in the following situation:
of finitely generated projective A-modules and a natural isomorphism of functors on the category of
The base change map is an isomorphism for a quasi-coherent sheaf
This map is a quasi-isomorphism provided that the following conditions are satisfied:[8] One advantage of this formulation is that the flatness hypothesis has been weakened.
However, making concrete computations of the cohomology of the left- and right-hand sides now requires the Grothendieck spectral sequence.
are affine (with the notation as above), the homotopy pullback is given by the derived tensor product Then, assuming that the schemes (or, more generally, derived schemes) involved are quasi-compact and quasi-separated, the natural transformation is a quasi-isomorphism for any quasi-coherent sheaf, or more generally a complex of quasi-coherent sheaves.
[9] The afore-mentioned flat base change result is in fact a special case since for g flat the homotopy pullback (which is locally given by a derived tensor product) agrees with the ordinary pullback (locally given by the underived tensor product), and since the pullback along the flat maps g and g' are automatically derived (i.e.,
The auxiliary assumptions related to the Tor-independence or Tor-amplitude in the preceding base change theorem also become unnecessary.
In the above form, base change has been extended by Ben-Zvi, Francis & Nadler (2010) to the situation where X, S, and S' are (possibly derived) stacks, provided that the map f is a perfect map (which includes the case that f is a quasi-compact, quasi-separated map of schemes, but also includes more general stacks, such as the classifying stack BG of an algebraic group in characteristic zero).
[10] The theorem on formal functions is a variant of the proper base change, where the pullback is replaced by a completion operation.
The see-saw principle and the theorem of the cube, which are foundational facts in the theory of abelian varieties, are a consequence of proper base change.
[11] A base-change also holds for D-modules: if X, S, X', and S' are smooth varieties (but f and g need not be flat or proper etc.
denote the inverse and direct image functors for D-modules.
are finite in each of the following cases: Under additional assumptions, Deninger (1988) extended the proper base change theorem to non-torsion étale sheaves.
In close analogy to the topological situation mentioned above, the base change map for an open immersion f, is not usually an isomorphism.
satisfies an isomorphism This fact and the proper base change suggest to define the direct image functor with compact support for a map f by where
is a compactification of f, i.e., a factorization into an open immersion followed by a proper map.
The proper base change theorem is needed to show that this is well-defined, i.e., independent (up to isomorphism) of the choice of the compactification.
Moreover, again in analogy to the case of sheaves on a topological space, a base change formula for