Can construct elements in of number fields, curves, products of curves. For ordinary Chow groups see e.g. Pirola (?) and Lewis.
Extend def to schemes over a ring:
Paris group: motivic cohomology and Arakelov theory, involving (I think) Cisinski, Deglise, Wildeshaus, Maillot, Rossler
Possible example for computation: Toric varieties (unless they give )
Possible thesis: Levine’s stuff + toric examples
Can visit Burgos after Christmas if useful
Formal properties wanted:
Q: What cat to work in?
A: Ultimately arbitrary arithmetic schemes (i.e. flat). To simplify, can assume smoothness/regularity (maybe of complex points of generic fiber). Easy start: Regular schemes.
Gillet comments: Might be able to reformulate conjectures on L-functions, or do computations related to polylogs.
Dear Professor Burgos,
Thanks for taking the time to talk to me in Toronto about the higher arithmetic Chow groups. I just wanted to write to you and ask you about what you said about Levine’s comments on the problem. You mentioned some email he had sent you related to an approach for regular schemes - would it be possible for you to forward the relevant parts of the email to me? Also, I think (but I might be wrong) that you mentioned some paper of Levine, but I am not sure which one you meant.
I look forward to further discussions with you about these problems! Again, thanks a lot for your interest and your time.
Best regards,
Andreas Holmstrom
Dear Andreas,
Below there is a copy of the emails I received from Levine. Not very informative but they suggest that it should be possible to extend the definition to varieties that are smooth over a dedekind domain after tensoring with \Q and there I hope one should be capable of make some computations. Besides the book on motives, te recent paper
Also has a discussion on moving lemma. Again over a field. But it micght be a starting point to extend the results to smooth varieties over a Dedekind domain.
email 1:
The definition I gave of cycle complexes for say a scheme of finite type over a Dedekind base-scheme B is exactly the same as Bloch’s definition if the base-scheme is semi-local, and in general one uses the Zariski hypercohomology of the presheaf (on B_\Zar) of cycle complexes. This satisfies a general localization theorem, but does not have a product or contravariant functoriality. If on the other hand one imposes an equi-dimensionality condition over B for the cycles and all their intersections with faces, this gives a subcomplex (or presheaf of subcomplexes over B_\Zar) which has the contravariant functoriality and products, but I can not prove the localization theorem for this subcomplex. If the base is a field, of course, the Chow’s moving lemma technique shows that the two are quasi-isomorphic, at least for quasi-projective schemes.
The functoriality for the equi-dimensional complex is discussed in
Chow’s moving lemma in homotopy theory. -theory {\bf 37} (1-2) 129-209 (2006).
email 2:
PS. I’m not sure if I’ve written this down somewhere, but the situation with \Q-coefficients is much better. One can use the argument I gave in “Bloch’s higher Chow groups revisited” to show that “ual“ cycle complexes, as well as the equi-dimensional cycle complexes, give higher algebraic K-theory after tensoring with \Q, at least for affine varieties. So, this shows that they are quasi-isomorphic after \otimes\Q for affine varieties. This in turn (relying on localization/Mayer-Vietoris) gives the full cycle complexes products and contravariant functoriality, in the derived category.
Best,
email 3:
Happy to help out. The argument of Bloch is in fact correct, once one has corrected all the moving lemma mistakes. I give a sketch of his argument with perhaps a few more details in my book Mixed Motives, Chap III, section 3.6, pg. 179-189. Although the notation in my book is pretty bad, I think that this particular section is self-contained, doesn’t rely on any categorical constructions and does the job. Please feel free to ask me about any details that are unclear.
nLab page on Burgos