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:
Contents
- Introduction
- Definition of the base change map
- Proper base change
- Direct image with compact support
- Flat base change
- Flat base change in the derived category
- Base change in derived algebraic geometry
- Variants and applications
- Base change for tale sheaves
- Applications
- References
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.
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
Here the subscript indicates the forgetful functor, i.e.,
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
This map exists without any assumptions on the maps f and g. It is constructed as follows: since
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
The above-mentioned introductory example is a special case of this, namely for the affine spectra
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
where
Proper base change
If X is a Hausdorff topological space, S is a locally compact Hausdorff space and f is universally closed (i.e.,
is an isomorphism. Indeed, we have: for
and so for
To encode all individual higher derived functors of
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).
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
To obtain a base-change result, the functor
In general, there is a map
Proper base change
Proper base change theorems for quasi-coherent sheaves apply in the following situation:
As the stalk of the sheaf
These statements are proved using the following fact, where in addition to the above assumptions
on the category of
Flat base change
The base change map
is an isomorphism for a quasi-coherent sheaf
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
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
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.,
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).
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
Base change for étale sheaves
For étale torsion sheaves
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
Under additional assumptions, Deninger (1988) 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
This fact and the proper base change suggest to define the direct image functor with compact support for a map f by
where
For the structural map
Similar ideas are also used to construct an analogue of the functor