Base change theorems

In mathematics, the base change theorems relate the direct image and the pull-back of sheaves. 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 formula_3 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.
Introduction.
A simple base change phenomenon arises in commutative algebra when "A" is a commutative ring and "B" and "A' "are two "A"-algebras. Let formula_4. In this situation, given a "B"-module "M", there is an isomorphism (of "A' "-modules):
Here the subscript indicates the forgetful functor, i.e., formula_6 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.
Definition of the base change map.
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 Cartesian square and formula_3 is a sheaf on "X". Here formula_11 denotes the higher direct image of formula_12 under "f", i.e., the derived functor of the direct image (also known as pushforward) functor formula_13.
This map exists without any assumptions on the maps "f" and "g". It is constructed as follows: since formula_14 is left adjoint to formula_15, 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 formula_20 and formula_21 finally yields the desired map.
The above-mentioned introductory example is a special case of this, namely for the affine schemes formula_22 and, consequently, formula_23, and the quasi-coherent sheaf formula_24 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 formula_25 at a time. In fact, similar arguments as above yield a map in the derived category of sheaves on "S':"
where formula_27 denotes the (total) derived functor of formula_13.
General topology.
Proper base change.
If "X" is a Hausdorff topological space, "S" is a locally compact Hausdorff space and "f" is universally closed (i.e., formula_29 is a closed map for any continuous map formula_30), then
the base change map
is an isomorphism. Indeed, we have: for formula_32,
and so for formula_34
To encode all individual higher derived functors of formula_13 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 .
 has extended the above theorem to non-abelian sheaf cohomology, i.e., sheaves taking values in simplicial sets (as opposed to abelian groups).
Direct image with compact support.
If the map "f" is not closed, the base change map need not be an isomorphism, as the following example shows (the maps are the standard inclusions) :
One the one hand formula_39 is always zero, but if formula_12 is a local system on formula_41 corresponding to a representation of the fundamental group formula_42 (which is isomorphic to Z), then formula_43 can be computed as the invariants of the monodromy action of formula_44 on the stalk formula_45 (for any formula_46), which need not vanish.
To obtain a base-change result, the functor formula_13 (or its derived functor) has to be replaced by the direct image with compact support formula_48. For example, if formula_49 is the inclusion of an open subset, such as in the above example, formula_50 is the extension by zero, i.e., its stalks are given by
In general, there is a map formula_52, which is a quasi-isomorphism if "f" is proper, but not in general. The proper base change theorem mentioned above has the following generalization: there is a quasi-isomorphism
Base change for quasi-coherent sheaves.
Proper base change.
"Proper base change theorems" for quasi-coherent sheaves apply in the following situation: formula_49 is a proper morphism between noetherian schemes, and formula_3 is a coherent sheaf which is flat over "S" (i.e., formula_45 is flat over formula_57). In this situation, the following statements hold:
As the stalk of the sheaf formula_73 is closely related to the cohomology of the fiber of the point under "f", this statement is paraphrased by saying that "cohomology commutes with base extension".
These statements are proved using the following fact, where in addition to the above assumptions formula_74: there is a finite complex formula_75 of finitely generated projective "A"-modules and a natural isomorphism of functors
on the category of formula_77-algebras.
Flat base change.
The base change map
is an isomorphism for a quasi-coherent sheaf formula_12 (on formula_80), provided that the map formula_81 is "flat" (together with a number of technical conditions: "f" needs to be a separated morphism of finite type, the schemes involved need to be Noetherian).
Flat base change in the derived category.
A far reaching extension of flat base change is possible when considering the base change map
in the derived category of sheaves on "S'," similarly as mentioned above. Here formula_83 is the (total) derived functor of the pullback of formula_84-modules (because formula_85 involves a tensor product, formula_20 is not exact when is not flat and therefore is not equal to its derived functor formula_83).
This map is a quasi-isomorphism provided that the following conditions are satisfied:
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.
Base change in derived algebraic geometry.
Derived algebraic geometry provides a means to drop the flatness assumption, provided that the pullback formula_119 is replaced by the homotopy pullback. In the easiest case when "X", "S", and formula_95 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.
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., formula_123). 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 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).
Variants and applications.
Proper base change also holds in the context of complex manifolds.
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.
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.), there is a quasi-isomorphism
where formula_125 and formula_126 denote the inverse and direct image functors for "D"-modules.
Base change for étale sheaves.
For étale torsion sheaves formula_12, there are two base change results referred to as "proper" and "smooth base change", respectively: base change holds if formula_128 is proper. It also holds if "g" is smooth, provided that "f" is quasi-compact and provided that the torsion of formula_12 is prime to the characteristic of the residue fields of "X".
Closely related to proper base change is the following fact (the two theorems are usually proved simultaneously): let "X" be a variety over a separably closed field and formula_3 a constructible sheaf on formula_131. Then formula_132 are finite in each of the following cases:
Under additional assumptions, extended the proper base change theorem to non-torsion étale sheaves.
Applications.
In close analogy to the topological situation mentioned above, the base change map for an open immersion "f", 
is not usually an isomorphism. Instead the extension by zero functor formula_135 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 formula_138 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 formula_21 vs. formula_48 does hold for non-proper maps "f".
For the structural map formula_141 of a scheme over a field "k", the individual cohomologies of formula_142, denoted by formula_143 referred to as cohomology with compact support. It is an important variant of usual étale cohomology.
Similar ideas are also used to construct an analogue of the functor formula_48 in A1-homotopy theory.