Where we’re at, and where we’re going
Over the course of the next 3 months I’m going to learn the basics of motivic cohomology (and the construction of Voevodsky’s geometric motives). During this time, I’ll make it a habit to make a blog post covering what I’ve learned each week. The goal will be, at the end of this semester, to have a concise overview of the theory in the form of ~12 blog posts. There will be four parts: I. Background, II. Categories of Motives, III. Motivic Cohomology, IV. Comparison results and computations.
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 the 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.
Warning: Regarding motivation and historical accuracies be very wary – as I probably have not checked any of the sources to repeat this. Be also wary of the maths but, to a lesser extent as I will double check this. By convention, all my schemes are separated and of finite type over a field.
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, respecting inclusions into summands, and 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). But the information is that of a contravariant functor from the category of open sets of some topological space to some other category, most likely something algebraic. In our case, a presheaf with transfers is not going to give rise to a sheaf on a topological space; it is going to give rise to a sheaf for each smooth scheme defined over .
The additional “with transfers” is essentially perpendicular to the above discussion. Recall we have a faithful embedding . If we have a presheaf with transfers , i.e. 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
which 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 isn’t if you’ve gotten this far in algebraic geometry) then you may be asking why are we considering a new category, again. The answer is simple, this one is abelian! It isn’t really difficult to show this (or we could refer to 1.6.4 in [Wei]) and comes down to defining the kernel and cokernel of a morphism of presheaves with transfers, say . But we can just define these to be, for every object , and similarly replacing by .
What is less obvious is knowing 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 using some category theoretic results 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. For this we will need the notion of a Grothendieck topology, . There are three 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 : it suffices to prove, for any fixed i, the restriction map is injective. 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 . 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. Using the functor of points it becomes a manipulation of set definitions to check these are, in fact, equal.
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 it is irreducible. This implies the generic fiber of the projection is dense in . Hence if we have two correspondences which agree on then, as is an open subset of an irreducible scheme it contains the generic point of , both are closed integral subschemes agreeing on a dense subset and are hence equal.
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 definition of the fibered product.
To see the exactness 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). Since maps to , we find for each pair i,j. (To be continued – I think this argument should be simpler than the one given in [MVW] so I’ll wait for a better proof to present itself).
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 mentioned above for 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
In the above the correspondence is the graph of the morphism deleting the th component.
It is a resolution of as an étale/Nisnevich sheaf by étale/Nisnevich sheaves respectively.
Proof. That these objects are sheaves is immediate after proposition 2.7. 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 to treat our objects as sheaves and compute homology in the category of sheaves. That is, we can check the complex 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.
Remark: It seems ironic that I don’t want to call a point, even though that is what it is, because I’ve been saying “points” only for the functor language so far. In this case I actually do mean an element of the underlying set of the scheme . I think fitting terminology would be to call it an “absolute point”. I might do that, but later; as I will finish the proof later.
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.