Thursday, November 29th, 2012 | Author: Konrad Voelkel
I'd like to compile a
short list of definitions of Weil and Cartier Divisors, Line Bundles and Invertible Sheaves, Class Groups and Picard Groups, Cohomology, (higher) Chow Groups and K-theory for algebraic schemes and their relations. I intentionally omit proofs, but there are some ideas. I couldn't resist to jot down some properties of the objects which are important to me (homotopy invariance, existence of pullbacks and pushforwards).
Let be an algebraic scheme over a field , i.e. a scheme with structural morphism of finite type. Whenever any coefficients appear, I chose , but it might be much more convenient to take in applications. It might also be quite convenient to restrict attention to varieties which are smooth over and assume perfect (since these conditions imply regular and normal), but I try to make these assumptions only when necessary. In fact, after writing this, I wish I would have stuck to rational coefficients and smooth schemes.
Cycles and divisors
A -dimensional cycle of is a -linear combination of -dimensional closed subvarieties, i.e. closed immersions from integral algebraic schemes of dimension . The group of all -dimensional cycles is denoted . If is equidimensional, the group of all codimension cycles is denoted . I will discuss rational equivalence and Chow groups below.
For equidimensional and regular in codimension , a Weil divisor is a -linear combination of codimension subvarieties (which are also called prime divisors). A Weil divisor is called effective if the coefficients of the prime divisors are non-negative. To a non-zero rational function one can associate a Weil divisor , where the sum runs over all prime divisors. Weil Divisors of the form are called principal divisors. The group of divisors modulo principal divisors is called the divisor class group. I will discuss the relation with Chow groups below.
A Cartier Divisor is a global section of , where is the sheaf of rational functions on . A Cartier divisor is called principal divisor if it is in the image of the quotient map from , i.e. if it can be represented by a global non-zero rational function. The group of Cartier divisors modulo principal divisors is called Cartier class group. I will discuss the relation with cohomology below.
Invertible sheaves and bundles
An invertible sheaf is a coherent -module sheaf which is locally free of rank , i.e. an invertible sheaf is locally isomorphic to . For an invertible sheaf the dual -module is a -inverse via the evaluation isomorphism .
An algebraic line bundle is an algebraic vector bundle of rank , i.e. a morphism of schemes with zero section , scalar multiplication morphism and vector addition morphism that turn each fiber into a -vector space of dimension , and there exists a Zariski-covering of and a local trivialization of such that the corresponding glueing morphisms are -linear in each fiber. From a general argument identifying locally free -modules as sheaves of sections of algebraic vector bundles (which gives an equivalence of categories), algebraic line bundles are equivalent to invertible sheaves. The group of all line bundles modulo isomorphism (with group structure from ) is called Picard group .
Definitions of some fancy objects
The -th K-group of a ring is the Grothendieck ring of projective modules, i.e. the group of isomorphism classes of projective modules modulo the relation identifying a module with if there exists a short exact sequence (which doesn't necessarily split). The -th K-group is the Grothendieck ring of algebraic vector bundles. Since projective modules over a ring are, as category, equivalent to vector bundles over its spectrum, . There are various isomorphic definitions of higher algebraic K-theory of a scheme, but I won't write any of these down. For a ring one can take . There is a filtration on called the Adams filtration, or -filtration (coming from the -ring structure), with weight--subspace of just .
For equidimensional and smooth over , write for -linear combinations of codimension closed subvarieties of which intersect all faces (for ) properly, where is the affine standard simplex. These fit together to form a simplicial abelian group (with respect to intersection product) and the homotopy groups are called higher Chow groups. Obviously, .
Motivic cohomology are certain Ext-groups in Voevodsky's category of mixed motives (one can take that as definition): , where is the presheaf with transfers associated to .
After looking at the comparison theorems below, the properties noted here become a lot nicer on smooth schemes, by using the comparisons to get more pullback/pushforward homomorphisms.
- Cycles can be pulled back along flat morphisms of constant relative dimension, which preserves codimension, rational equivalence and is compatible with the intersection product. The same holds for Weil divisors. There is also flat pullback for higher Chow groups.
- Cartier divisors can be pulled back along any map whose image is not contained in the support of the divisor. By taking a linearly equivalent Cartier divisor, if necessary, one may always pull back the class of a Cartier divisor (on an integral scheme).
- Bundles can be pulled back to bundles along arbitrary morphisms and isomorphic bundles pull back to isomorphic bundles, hence there is a pullback on the Picard group. This generalizes to algebraic K-Theory.
- On sheaf cohomology one has arbitrary pullbacks (sheaf cohomology is a contravariant functor, after all) and the same is true for motivic cohomology.
- Cycles can be pushed forward along proper morphisms, which preserves dimension and rational equivalence. Since Weil divisors are codimension 1 cycles, they admit proper pushforward only along maps of relative dimension . There is the degree map, which is just pushforward along the structural morphism . I don't know whether higher Chow groups also have proper pushforward... do you?
- I don't know if there is any kind of general pushforward for Cartier divisors or classes thereof.
Bundles can be pushed forward (in the sense of pushforward of sheaves that yields a bundle again) along finite flat morphisms, and there are more morphisms whose pushforward is a vector bundle again. I don't know if there is a nice criterion for which kind of morphism allows pushforward in K-Theory.
- For cohomology, pushforwards of sheaves exist in general, but you don't get a pushforward morphism on the cohomology in general. If you take a proper map which is in some sense oriented, then one gets pushforwards on ordinary (singular) cohomology, which look like integration over the fiber in the de Rham picture. I don't know what the general statement for motivic cohomology looks like... do you?
- Chow groups have homotopy invariance, i.e. , where the isomorphism is induced by pullback along the projection . In particular, the Weil divisor class group has homotopy invariance. Even the higher Chow groups are homotopy invariant, i.e. .
- The Cartier class group is not homotopy invariant in general, but I don't know good examples.
- The Picard group (and the K-Theory) are homotopy invariant on regular schemes (but I don't know counter-examples in general).
- Sheaf cohomology is not homotopy-invariant in general, but if you take a locally constant sheaf as coefficients, it is. However, in this article, coefficients are most relevant, and there we don't have homotopy invariance. Motivic cohomology, on the other hand, is homotopy invariant (by construction).
As far as I can see, all these comparisons are compatible with pullbacks (as far as they exist). For pushforwards, the precise relationship between the Chern character and pushforwards is called Grothendieck-Riemann-Roch theorem.
Weil divisor class group and Chow group of codimension 1 cycles
The usual definition of rational equivalence (that shows up in the definition of the Chow group as in the book of Y.André on motives) looks rather different from the usual definition of linear equivalence (that shows up in the definition of the Weil divisor class group). Linear equivalence is just equality modulo adding principal divisors.
Rational equivalence of two cycles of codimension in is the existence of a cycle of codimension in such that the projection to is dominant, and .
These two equivalence relations on codimension-1 cycles agree. From linear equivalence you can easily cook up a joining the two divisors, but the other direction is tricky. One has to relate the pushforward in the latter definition to the former. The crucial technical result (Proposition 1.4b in Fulton's book on intersection theory) states that for a proper surjective morphism of varieties of the same dimension and a non-zero rational function, , where is the norm of the field extension . This is applied in the situation where we have as in the definition of rational equivalence, but consists of just a subvariety (the general case is done by extending linearly), so there is a morphism which is induced by the projection (so, determines a rational function ) and the other projection induces morphisms from the scheme-theoretic fiber for any rational point , which are isomorphisms onto some subscheme we want to call . Then one has , hence , which is linearly equivalent to (since it's a principal divisor).
Cartier class group and cohomology:
The short exact sequence of sheaves yields a long exact cohomology sequence which reads and identifies .
Line bundles and cohomology:
Given a line bundle one can take any local trivialization over an open cover with patching data and these patching data form a Cech 1-cocycle over with values in , thus a class in Cech cohomology . In the limit, one gets a class in , thus from general nonsense a class in the sheaf cohomology group . Different trivializations yield cohomologous cocycles, hence the same Cech classes.
It also works the other way around: Take such a cohomology class, represent it by a Cech 1-cocycle over some open cover (which has to be fine enough) and write down the bundle patched together from trivial bundles via the patching data. Taking a different 1-cocycle in the same class (or a different open cover) yields an isomorphic bundle, hence .
Cartier divisors and line bundles:
For each Cartier divisor we can choose an open affine cover such that is represented by sections of over . Then we define a line bundle as the sub--module of generated by the over . This gives a monomorphism .
To any line bundle with embedding we can associate a Cartier divisor such that , by taking to be the inverse of a local generator of over (where the have to be a trivializing cover). This is obviously an inverse to the other construction.
If is integral, , a constant sheaf, and then we can embed every line bundle in by , since is locally constant, hence constant. Therefore, on integral, .
Compatibility of the isomorphisms so far:
The isomorphism class of line bundles we attached to a Cartier class turns out to be the same isomorphism class of line bundles specified by the cohomological data given by the Cartier class, which one can see explicitly by taking a Cartier divisor and assign the isomorphism class of line bundles specified by the patching data .
Cartier divisors and Weil divisors:
On integral, separated, noetherian and regular in codimension , Cartier divisors can be mapped to locally principal Weil divisors, by representing the Cartier divisor over a cover as some and for each prime divisor taking some index such that , then is well-defined (doesn't depend on ) and is a locally principal Weil divisor.
If is furthermore normal, each locally principal Weil divisor comes from a Cartier divisor, since for each point we can take the restriction , a divisor on , and is a UFD, so is a principal divisor, and defines a divisor on which restricts to as well, so agrees on an open neighborhood with ; the various give a Cartier divisor, since is normal, so on any open with inducing the same Weil principal divisor, is a section of .
If is furthermore locally factorial, every Weil divisor is locally principal, so . Since principal divisors agree, . In particular this holds for smooth .
The map from an isomorphism class of line bundles to a linear equivalence class of Cartier divisors to a rational equivalence class of Weil divisors is also called , the first Chern class. Note how one has such a map even if and are only monomorphisms. This map is called , the first Chern class of a line bundle.
I want to summarize, for smooth over , we have
Comparison of fancy objects
Comparison of Chow groups and K-Theory in general, for smooth over :
where the indicates the -filtration.
This is the Chern character, which you can build by identifying with the -group of coherent sheaves on (there smoothness is used), and then one can take a resolution of a coherent sheaf by vector bundles (on a stratification) and define ordinary chern classes for vector bundles via a splitting principle and .
This generalizes to higher Chow groups and higher K-Theory (but I don't know who proved that):
Voevodsky proved that these are strongly related to motivic cohomology:
for smooth over and a perfect field, where the map is the so-called motivic cycle class map.
We recover the comparison of divisors with line bundles and cohomology (but now with rational coefficients):
where is just and and .
I don't know how to prove that stuff :-) but I hope I'll learn that some day.