Where we’re at, and where we’re going
We last left off with: Motivic Cohomology I: correspondences. In this post it was described how we could construct a new category, , so that we have a faithful embedding of the category of smooth schemes over into this newly defined category. The objects remained the same, but we now have correspondences as morphisms. It was observed is an additive category and, in addition, has a symmetric monoidal structure given by the tensor product.
In this post, we’ll construct an abelian category where embedds fully faithfully through the use of presheaves and Yoneda’s lemma. Afterwards, we extend the sheaf notion to our new presheaves for various Grothendieck topologies. We show our representable objects form sheaves in these respective topologies. We conclude by observing, for the étale and Nisnevich topologies, these presheaves admit a sheafification functor.
Presheaves With Transfers
We’ll start out with the definition:
(2.1) Definition: A presheaf with transfers is a contravariant additive functor from the category to the category of abelian groups .
The condition that a presheaf with transfers be an additive functor can be reformatted into conditions (C1) an (C2) below.
(C1) a presheaf with transfers is a contravariant functor taking sums to sums, taking summands with their inclusion into a sum to summands with their inclusion into a sum, and taking the zero object to the zero object.
(C2) for every smooth schemes and presheaf with transfers , the map of sets associated to the functor is a homomorphism of abelian groups which commutes with the composition maps and .
Our interest will be in a new category, , whose objects are presheaves with transfers and whose morphisms are natural transformations between functors.
Remark: I’m not overly fond of the use of the term presheaf with transfers. The concept of a sheaf outdates the concept of a presheaf. The word sheaf (on a topological space) was used because it gives some intuitive feel for what a sheaf is (originally a sheaf was defined as the espace étale associated to a presheaf). The terminology “presheaf” implies, at least I interpret it as, information giving rise to a sheaf (on a topological space, via sheafification). The information contained in a presheaf is that of a contravariant functor from the category of open sets of some topological space to some other category. In our case, a presheaf with transfers is not going to give rise to a sheaf on a specific topological space. A presheaf with transfers is going to give rise to a sheaf on every smooth scheme defined over . Maybe pluri-sheaf, or multi-sheaf, would have been more appropriate.
The additional terminology “with transfers” is essentially perpendicular to the above discussion. Recall we have a faithful embedding . If we have a presheaf with transfers , in other words a contravariant additive functor , we can consider the composition as a restriction of a presheaf to the category of smooth schemes. That is to say, we have a forgetful functor
and transfers are exactly the data which we forget.
This can be phrased another way. Since in we have , any presheaf with transfers will associate to a given correspondence a homomorphism of abelian groups like so
This association induces a map on sets . Condition (C2) then guarantees this induces a homomorphism of abelian groups . Transfers are, in this wording, those maps associated to correspondences which do not arise as the graphs of honest morphisms (my slang term honest means a morphism in the category ).
If this is your first time working with categories of functors (it probably isn’t) then you may be asking why are we considering a new category. Again. The answer is simple, this one is abelian! It isn’t that difficult to show this (or we could refer to 1.6.4 in [Wei]) and the proof comes down to defining the kernel and cokernel of a morphism of presheaves with transfers, say . But, we can just define the kernel to be, for every object , and similarly define the cokernel replacing by .
What is less obvious is knowing that having enough inejctives and projectives implies
(2.2) Proposition: The category has enough injectives and projectives.
Reference. [Wei] exercises 2.3.7, 2.3.8.
Essentially, we can theoretically compute cohomology (I say theoretically compute but, a better phrase is probably “there exists a satisfactory cohomology theory satisfying various functorial properties”) for objects in the category .
For now, we’ll leave this category here, because we’ll actually be using a different category, constructed from by applying some category-theoretic-tools which I will cover in my next post. However, we’ll point out that Yoneda’s lemma shows
(2.3) Proposition: There is a fully faithful embedding .
If is a smooth scheme, then the embedding takes to the functor , or the functor which assigns to any scheme the abelian group . (We’ll use and interchangably). The functors are said to be represented by . In the -linear category we’ll call the functor represented by (when is considered as an object in ) instead of . We can extend all of the above definitions to the category . We write for the category of presheaves .
Remark: We can assign a similar functor to any scheme smooth or not. That is, we can also define . However, we can’t say much else about this functor.
Sheaves with transfers
Now let us specialize to the concept of sheaves with transfers. To have a meaningful discussion of these objects we will need the notion of a Grothendieck topology, . There are three topologies we focus on but, we can use any topology which satisfies the following condition:
(T1) For any smooth scheme , the category consisting of covers of and allowed morphisms of covers includes naturally into (schemes over ). We require the the inclusion factors .
What I want from this condition is that all of our covers are actually smooth schemes as well. This way we can use the information we have about the topology for a fixed without having to reprove it for just smooth covers. In the three topologies we focus (Zariski, Nisnevich, and Étale) this won’t be a problem since:
(2.4) Proposition: Let be a smooth scheme. Any topology which occurs as a subcategory of , the étale topology on , satisfies condition (T1).
Proof. Since étale morphisms are smooth, the composition of any cover with the smooth structure map is smooth.
We can expand the sheaf condition from one scheme to with:
(2.5) Definition: Let be a Grothendieck topology satisfying (T1). A presheaf of abelian groups is a sheaf for the topology if the restriction to is a sheaf for every smooth scheme .
This is equivalent to the restriction satisfying exactness of the usual sequence for every cover and . In the cases we are concerned with (Zariski, étale, and Nishnevich topologies), it suffices to consider only finite indexing sets (that is, we need only consider covers which consist of finitely many covering maps). This is because, for a Noetherian site, there is a natural equivalence between the categories which contain only finite covering maps and the general case (a Noetherian site is one for which any cover has a finite subcover – in our case this follows from the fact étale maps are open, and any smooth scheme being finite type over is quasi-compact). For this result, which we use in the proof of proposition 2.7, we refer to [Mil], Chapter 3 section 3 proposition 3.5 page 112.
We extend the definition to similarly:
(2.6) Definition: A presheaf with transfers will be called a sheaf with transfers for the topology, if it’s restriction to is a sheaf for the topology.
The subcategories consisting of Zariski, étale, and Nisnevich sheaves with transfers will be denoted , , and respectively. For the most part, we’ll prove results in the category of sheaves for the étale and Nisnevich topologies (although we may have to digress to the cdh topology, among others, if we want to extend our definition of motives to all schemes). The Zariski topology will be our place to define most objects, like motivic cohomology. We’ll use the results we obtain in these finer topologies to prove more general results which we then specialize to the Zariski topology.
(2.7) Propsoition: For any scheme , the presheaf with transfers is a sheaf with transfers for the Zariski, étale, and Nisnevich topologies.
Proof. We’ll check exactness of the sequence
for any cover .
Injectivity of : Unraveling definitions we need to show is injective. Let be the group of all cycles in the product . By definition we have inclusions . For the étale (and hence Zariski and Nisnevich) topology, the map is flat and from intersection theory we know the flat pullback of cycles induces an injective map (to see the injectivity, note that the map is given by taking a cycle to its preimage; since the cover is required to surject onto , every cycle has a preimage whose image is itself). Thus, the result follows if we can show the following diagram commutes:
where the top horizontal arrow is composition , the bottom horizontal arrow is flat pullback of cycles, and the vertical arrows are the inclusions. These agree by construction.
Remark: For the Zariski topology one can make this more geometric: assume is connected. Then any elementary correspondence is surjective over and, since is smooth, is irreducible. This implies the generic fiber of the projection is dense in . Hence if we have two correspondences which agree on for some dense open then, as is an open subset of an irreducible scheme, it contains the generic point of . Hence, both are closed integral subschemes agreeing on a dense open subset of and are therefore equal on all of . Note also that this implies if then the restriction is injective.
Following the definitions of the maps , it’s clear this is a complex since
Here we used the fact that we have equality of maps by the construction of the fibered product.
Exactness in the middle: suppose . We’ll invoke the proposition of Milne mentioned earlier to assume k is finite, or that our covering was finite all along (since we haven’t used it until this point). This lets us prove the proposition by working with only two indices .
So assume we have a pair so that . We want to show that is in the image of . Essentially what we need to do is find a way to extend correspondences on the “intersection” to . We can do this by using a “generic” argument or, by reducing the proof to the case is field.
To do this reduction, we first extend everything to the function field, call it , of (if isn’t irreducible, then we first replace it by an irreducible component and then argue by additivity of ). Next, we take limits over Zariski open covers of . Since are étale over their restrictions over the opens of are as well. This means, in the limit, we are left with two étale covers of which we know to be just a product of finite separable extensions of . Call and the product of fields we get in the inverse limits of and respectively. If we’re able to show the proposition when is a field, then we’ve shown the proposition for the sequence
We proceed by a diagram chase using the commutative diagram below.
The bottom row of this diagram is exact and all of the vertical arrows in this diagram are injective by the remark above. So, to show the claim we could try to verify the map is surjective. But this is not true! (For example, the hyperbola defines a finite correspondence in that can’t come from a finite correspondence in . In other words, the inclusions are strict). Instead, from the exactness of the bottom row in the above diagram, we get a Zariski open subset so that the sequence restricted to is exact:
In particular, we find a correspondence so that under the second arrow from the left. But this is good enough: we can assume are elementary correspondences (or we can work with each summand) to argue that, since that in fact there is an equality between the closure of the restriction of , , and . The same argument works for , which proves the proposition (modulo the next paragraph).
It remains to prove the proposition when is a field. So we can assume also that are a product of finite separable extensions of . We can assume further are fields themselves (by the additivity of the functor) and that for some field by extending to their composite. By the primitive element theorem we can write for some irreducible -polynomial which completely splits over . Factoring over shows our correspondences restrict to a disjoint union of finitely many indexed by elements of the Galois group . Now, if we have a pair which is mapped to then for every or, in other words, is invariant under . But this means exactly that it comes from an element of which completes the proof.
For a cover we can construct a complex, the Čech complex, and denote it . In both the étale and Nisnevich topologies this complex is, in fact, a resolution by sheaves in the respective topology. In the Zariski topology this fails, and we’ll have to replace it by something different. Eventually we’ll show how to do this but, for now we focus on doing it for the cases present and, after learning why we were doing it in the first place some posts later, we’ll come back to the complicated Zariski discussion.
(2.8) Proposition: The Čech complex associated to a covering map (for example any finite cover gives rise to such a covering map in the étale and Nisnevich topologies by taking to be the disjoint union of the ) is defined as the complex
Here the correspondence is the graph of the morphism deleting the th component. The Čech complex is a resolution of as an étale/Nisnevich sheaf by étale/Nisnevich sheaves respectively.
Proof. Showing this is a complex amounts to showing equality of maps in the category . As these are presheaves, we just need to compute the composition of two terms starting from an arbitrary scheme and a correspondence . Explicitly our maps are:
What we can do is: see how this double sum behaves on points , using the notation to represent the map . If we did this, we’d find the same cancellation that occurs in the proof that, say, the boundary map in singular homology is actually a differential.
Remark: We can do this because these maps actually behave this way for -valued points, in the functor language.
To see the comlpex has no homology, it suffices, since the objects of this complex are sheaves by Proposition 2.7 above, to compute homology in the category of sheaves. That is to say, we can check this complex is exact at the level of stalks. Or, for every smooth scheme and every , the morphism will be exact if it is when evaluated on in the étale topology, and in the Nisnevich topology. The idea will be to first rewrite the complex in a different, but equivalent, form and then use Hensel’s lemma to define a chain homotopy between the identity on this complex and the 0-map on this complex, which will prove the theorem by general homological algebra. The actual difficulty is finding a way to employ Hensel’s lemma (and of course checking that we can choose lifts up to homotopy).
Unfortunately, we need to introduce some notation for clarity. Let be as above, let be an arbitrary smooth scheme, and let be a closed subscheme of which is finite and surjective over . denote the free abelian group of closed irreducible subschemes of which are finite and surjective over . To clarify how to think about this, note if then there’s an equality as groups; in general we have an equality where the limit runs over all closed subschemes . Using this, we can rewrite our complex as a direct limit of complexes:
The utility in doing this is that it’s possible to show, given a closed subscheme finite and surjective over , that we can define a map whenever is a Hensel local scheme (which we can also assume, since we’re working stalkwise). To see this, note that since is a Hensel local scheme and is finite over , then is also a Hensel local scheme. By [The Stacks Project, Lemma 10.148.3], from the projection map there is a splitting (i.e. a map with ). From this section we get maps . We define the chain homotopy by composing with as correspondences.
We still need to check that . In simple cases I can check this but I don’t know a general approach right now. (It can probably be done formally, in the same way we can check most of the other claims about compositions in this field).
Finally, we arrive at our last result for this section which concerns sheafification in the various topologies. Let be a Grothendieck topology satisfying (T1). Let the inclusion of the subcategory of sheaves for this topology into the category of presheaves be denoted . When there is an adjoint we say that admits sheafification.
When is either the étale or the Nisnevich topologies, it is known there exists a sheafification. We’ll denote this map and respectively. For any presheaf with transfers , we can restrict to a presheaf on via , and then sheafify with or . Let or be the sheaf on that we get after applying this procedure.
(2.9) Proposition: There is a presheaf with transfers (or for the Nisnevich topology) so that any morphism (resp. ) as presheaves on factors uniquely through (resp. ) with the first map being a morphism of presheaves with transfers and the latter a morphism of sheaves on the category .
Reference: [MVW] Theorem 6.17, page 42 in the étale case; Theorem 13.1, page 99 for the Nisnevich.
Remark: Again this proposition fails for the Zariski topology but, we can fix this by only considering presheaves with transfers satisfying additional conditions. We’ll take this approach in a later post.
Some arguments I won’t go into details about allow us to conclude
(2.10) Proposition: the categories and have enough injectives.
Reference: Proposition 13.1 in [MVW] for the Nisnevich case which really just says the proof in the étale case, Proposition 6.19 in [MVW], carries through to the Nisnevich topology as well.
Tensor Products and Internal Homs of Presheaves with Transfers
Our construction of the tensor product for presheaves with transfers is going to extend our definition of the tensor product in the category . Recall we defined, for any smooth schemes over , . We can, in fact, do this construction with coefficients for some ring .
Recall the Yoneda embedding
defined on objects as . Using the Yoneda lemma this immediately gives:
(2.11) Lemma: The representable objects are projective objects of .
Proof. The Yoneda lemma says for an arbitrary functor . Hence, if we have an exact sequence of presheaves
is exact since it is canonically the sequence
which is exact by assumption..
From (2.11) we can also deduce
(2.12) Lemma: Every presheaf has a projective resolution. That is, has enough projectives.
Proof. This is a result of the natural surjection, for arbitrary presheaves ,
From here the resolution can be constructed inductively, at each step having the natural surjection onto the kernel of the previous map.
Remark: In the above, it’s implicitly used that the objects of form a set instead of a proper class.
Lemma (2.11) and (2.12) have the benefit of allowing us to define a tensor product structure extending the one we’ve already defined on . Note, if we try to naively define then we lose additivity (i.e. for a smooth connected scheme we would have which is not generally equal to ) , hence this is not a presheaf with transfers.
Instead, we proceed as follows.
(2.13) Construction: From the Yoneda embedding we can define a product on projective objects: which has the same formal properties as on .
We extend the definition of by defining, for arbitrary direct sums .
Now let be arbitrary presheaves with transfers, and projective resolutions as given in (2.12). The total complex is defined by the previous two pargraphs.
(2.14) Definition: We write for the total complex . The tensor product of presheaves with transfers and internal Hom presheaf with transfers are defined:
Remark: These are well-defined up to chain homotopy equivalence since any projective resolutions of the same functor are homotopy equivalent.
(2.14) Corollary: .
Proof. Since are projective, they form a projective resolution of length 1. The first equality is then clear. The last equality follows from the way we extended the product via the Yoneda embedding (2.13).
The internal Hom and tensor product also satisfy expected exactness properties.
(2.15) Lemma: The functor is right adjoint to . Hence, is left exact and is right exact.
Proof. We have
[Mil] Etale Cohomology – James Milne. Not to be confused with his later work, “Lectures on Etale Cohomology”. Princeton Mathematical Series.
[MVW] Motivic Cohomology – Mazza, Voevodsky, Weibel. Link. (Try Google if the link expires).
[Wei] An Introduction to Homological Algebra – Weibel. Link. Cambridge Studies in Advanced Mathematics. Or Google it.