Dear Professor Scholl,
Thanks for your email. Am now back to work, and will be around for the whole of Michaelmas term, except for the week in Toronto at the Arakelov workshop (by the way, I got money from them which appears to cover all expenses).
I haven’t been working very well lately, but I have some things I would like to discuss. In addition to the questions below, I have some general issues about how to find meaningful small “exercise-like” problems. The thesis of Feliu is very nice, but there is no obvious (to me) open questions accessible enough to just start working on.
Questions:
What can one say about the following situation (please tell me even obvious and trivial things): Suppose C is a subcat of “the cat of mixed motives over Q”. For example the cat of one-motives, or the cat obtained from MM_Q by imposing integrality at one place, at all finite places, or at all places, or the category of mixed realizations, or the mixed motives (realizations) coming from simplicial varieties, etc. Questions: 1: What can one say about how the Ext groups Ext^i_C(M,N) varies with the category C - which categories would (be expected to) give the same answer etc? 2: To what extent can these various Ext groups be “computed” in some sense? This is still very mysterious to me and I have no intuition whatsoever. Which of them should vanish?
You mentioned earlier the possibility of constructing a Voevodsky-style triangulated category which would give rise to higher Arakelov Chow groups in the same way as Voevodsky’s category gives motivic cohomology. Would you expect this triangulated category to be the same as the (bounded???) derived category of the subcategory of MM_Q which is obtained by imposing integrality at all places including the infinite place? If not, what would the relation be between these two categories?
Do you know what Feliu might mean by the term “arithmetic ring”?
When one defines mixed motives through realizations, it is done using the l-adic realizations, the Betti and the de Rham realizations. Is there any moral reason that this set of realizations is the right one? Could it be enough with a subset of these, or could there be a host of other realizations (cohomology theories) which one could/should add to the above?
Deligne describes/defines the cohomology of simplicial varieties, with mixed Hodge structures and a spectral sequence involving the cohomology of the components of the simplicial variety. Does a similar picture holds when talking about other cohomology theories? In general, which cohomology theories extend to simplicial schemes?
Best regards,
Andreas
Dear Andreas, These are interesting questions. I am back in the dept tomorrow and it would be good to discuss these then. But for now here are some clues.
The abelian category MM of mixed motives (which alas does not yet exist in the Voevodsky sense) would be generated by the motives of diagrams of smooth schemes. But I think one can do better than this, and for example it should be generated by motives of smooth quasi-projective schemes - or equally well by motives of projective (not necessarily smooth) schemes.
The category of 1-motives is a full subcategory of the Voevodsky category, so in particular of its hypothetical heart i.e. MM.
The integrality conditions are just conditions on the objects, so Hom=Hom and the map on Ext^1 is injective.
For number fields one expects all the motivic Ext^i to vanish for i>1, so that would be the end of the story.
I’ll talk to you later (I have to go to catch my plane!) - Tony
Dear Professor Scholl,
I will be in the department for the whole day tomorrow, and it would be great to meet if you have time.
What exactly do you mean by diagrams below? Finite diagrams? And what does one mean by the cohomology of a diagram of schemes?
The abelian category MM of mixed motives (which alas does not yet exist in the Voevodsky sense) would be generated by the motives of diagrams of smooth schemes.
Here is another related idea which I woke up with this morning (I haven’t thought of this in any detail, so it might of course be completely wrong for some trivial reason). Fix a base field. Let SV be the category of simplicial smooth projective varieties, and let Var be the category of all varieties, or maybe all smooth varieties. Take a cohomology theory on Var which is supposed to factor through the category of mixed motives. It would be nice if this cohomology functor factored though SV by Deligne’s construction, but this does not quite happen, if I understand things correctly. However, it “almost” happens, and I wonder if this “almost” can be explained as follows. Could it be that there is a model category structure on SV, and that the cohomology functor factors through the homotopy category Ho(SV)? Assuming that a model structure exists (will come back to this below), I have a vague feeling that this is what happens, because given two different simplicial resolutions, one can finds a third which maps to them both etc, and this kind of “zigzag” picture feels similar to descriptions of the homotopy category, so that two different simplicial resolutions would be isomorphic in Ho(SV).
Now if this holds for every (mixed Weil???) cohomology theory, then the functor from Var to MM, should also factor through Ho(SV), hence Ho(SV) behaves like a universal target for cohomology theories on Var (even “more universal” than MM). Could it be possible to formally “linearize” or “abelianize” the category Ho(SV) and actually get the abelian category of mixed motives???
Here is an ugly attempt to draw a diagram, where SmPr is smooth projective, and MM is mixed motives. There is (I guess) also a diagonal arrow from SV to MM which I cannot draw in this email, but which factors through Ho(SV), and the above would just say that there is a vertical arrow from Var to Ho(SV) which make both parts of the diagram commute.
SmPr —> Var —> MM
I I V
SV —> Ho(SV)
Now for this to make sense one needs a model category structure on SV, or at least a more general structure which allows us to make sense of Ho(SV). (In general, I would like to know: what are the conditions on a category C that makes the category of simplicial objects in C a model category?). Is this a well-known problem, and do you know the answer? If one thinks of a variety as a functor from rings (k-algebras) to sets, then a simplicial variety is a functor from k-algebras to simplicial sets. (This functor category (call it F) seems to me to be worth studying for its own sake, but I have not seen this done anywhere). Morally, I think a functor category from any category to a model category should inherit a model structure, although I think this is nontrivial (Thomason???). Hence one could believe in a model structure on F. Maybe this induces a model structure on the subcategory of F which is actually given by simplicial varieties - I guess one would have to check how the descent conditions relate to the model structure. This also looks like an interesting question in itself, regardless of the rest.
Does this make any sense? If it works, would it be of any use at all?
Andreas
nLab page on Mixed motives and Arakelov stuff fall 2008