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 is a sheaf on X.
Such theorems exist in different branches of geometry: for topological spaces and proper maps f, in algebraic geometry for 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. In this situation, given a B-module M, there is an isomorphism :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.
Definition of the base change map
The base change theorems presented below all assert that, that the following base change mapis an isomorphism, where
are continuous maps between topological spaces that form a Cartesian square and is a sheaf on X. Here denotes the higher direct image of under f, i.e., the derived functor of the direct image 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
and so
The Grothendieck spectral sequence then gives the first map and the last map 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.
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 at a time. In fact, similar arguments as above yield a map in the derived category of sheaves on S':
where denotes the derived functor of.
General topology
Proper base change
If X is a Hausdorff topological space, S is a locally compact Hausdorff space and f is universally closed, thenthe base change map
is an isomorphism. Indeed, we have: for,
and so for
To encode all individual higher derived functors of 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.
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 :One the one hand is always zero, but if is a local system on corresponding to a representation of the fundamental group , then can be computed as the invariants of the monodromy action of on the stalk , which need not vanish.
To obtain a base-change result, the functor has to be replaced by the direct image with compact support. For example, if is the inclusion of an open subset, such as in the above example, is the extension by zero, i.e., its stalks are given by
In general, there is a map, 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: is a proper morphism between noetherian schemes, and is a coherent sheaf which is flat over S. In this situation, the following statements hold:- "Semicontinuity theorem":
- * For each, the function is upper semicontinuous.
- * The function is locally constant, where denotes the Euler characteristic.
- "Grauert's theorem": if S is reduced and connected, then for each the following are equivalent
- * is constant.
- * is locally free and the natural map
- If, for some p, for all, then the natural map
These statements are proved using the following fact, where in addition to the above assumptions : there is a finite complex of finitely generated projective A-modules and a natural isomorphism of functors
on the category of -algebras.
Flat base change
The base change mapis an isomorphism for a quasi-coherent sheaf , provided that the map is flat.
Flat base change in the derived category
A far reaching extension of flat base change is possible when considering the base change mapin the derived category of sheaves on S', similarly as mentioned above. Here is the derived functor of the pullback of -modules.
This map is a quasi-isomorphism provided that the following conditions are satisfied:
- is quasi-compact and is quasi-compact and quasi-separated,
- is an object in, the bounded derived category of -modules, and its cohomology sheaves are quasi-coherent
- and are Tor-independent over, meaning that if and satisfy, then for all integers,
- One of the following conditions is satisfied:
- * has finite flat amplitude relative to, meaning that it is quasi-isomorphic in to a complex such that is -flat for all outside some bounded interval ; equivalently, there exists an interval such that for any complex in, one has for all outside ; or
- * has finite Tor-dimension, meaning that has finite flat amplitude relative to.
Base change in derived algebraic geometry
provides a means to drop the flatness assumption, provided that the pullback is replaced by the homotopy pullback. In the easiest case when X, S, and are affine, the homotopy pullback is given by the derived tensor productThen, assuming that the 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 agrees with the ordinary pullback, and since the pullback along the flat maps g and g' are automatically derived. 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 stacks, provided that the map f is a perfect map.
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, there is a quasi-isomorphism
where and denote the inverse and direct image functors for D-modules.
Base change for étale sheaves
For étale torsion sheaves, there are two base change results referred to as proper and smooth base change, respectively: base change holds if is proper. It also holds if g is smooth, provided that f is quasi-compact and provided that the torsion of is prime to the characteristic of the residue fields of X.Closely related to proper base change is the following fact : let X be a variety over a separably closed field and a constructible sheaf on. Then are finite in each of the following cases:
- X is complete, or
- has no p-torsion, where p is the characteristic of k.
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 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 of the choice of the compactification.
Moreover, again in analogy to the case of sheaves on a topological space, a base change formula for vs. does hold for non-proper maps f.
For the structural map of a scheme over a field k, the individual cohomologies of, denoted by 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 in A1-homotopy theory.