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:
is a Cartesian square 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.
- 1 Introduction
- 2 Definition of the base change map
- 3 General topology
- 4 Base change for quasi-coherent sheaves
- 5 Base change for étale sheaves
- 6 See also
- 7 Further reading
- 8 Notes
- 9 References
- 10 External links
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.
Definition of the base change map
|Image functors for sheaves|
|direct image f∗|
|inverse image f∗|
|direct image with compact support f!|
|exceptional inverse image Rf!|
|Base change theorems|
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 is a sheaf on X. Here denotes the higher direct image 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)
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.
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 (total) derived functor of .
Proper base change
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 Schnürer & Soergel (2016).
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 is always zero, but if is a local system on corresponding to a representation of the fundamental group (which is isomorphic to Z), then can be computed as the invariants of the monodromy action of on the stalk (for any ), which need not vanish.
To obtain a base-change result, the functor (or its derived 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 (i.e., is flat over ). In this situation, the following statements hold:
- "Semicontinuity theorem":
- If S is reduced and connected, then for each the following are equivalent
- is constant.
- is locally free and the natural map
- is an isomorphism for all .
- Furthermore, if these conditions hold, then the natural map
- is an isomorphism for all .
- If, for some p, for all , then the natural map
- is an isomorphism for all .
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 map
is an isomorphism for a quasi-coherent sheaf (on ), provided that the map 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 is the (total) derived functor of the pullback of -modules (because involves a tensor product, is not exact when g is not flat and therefore is not equal to its derived functor ). 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 (for example, could be a bounded complex of quasi-coherent sheaves)
- 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 .
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 is replaced by the homotopy pullback. In the easiest case when X, S, and 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., ). The auxiliary assumptions related to the Tor-independence or Tor-amplitude become unnecessary 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).
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.
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 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 (the two theorems are usually proved simultaneously): 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.
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,
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 vs. does hold for non-proper maps f.
- Grothendieck's relative point of view in algebraic geometry
- Change of base (disambiguation)
- Base change lifting of automorphic forms
- Esnault, H.; Kerz, M.; Wittenberg, O. (2016), A restriction isomorphism for cycles of relative dimension zero, arXiv:
- The roles of and are symmetric, and in some contexts (especially smooth base change) the more familiar formulation is the other one (dealing instead with the map for a sheaf on ). For consistency, the results in this article below are all stated for the same situation, namely the map ; but readers should be sure to check this against their expectations.
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- Iversen (1986), the four spaces are assumed to be locally compact and of finite dimension.
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