an X (by Bloch, Levine) as the Zariski-hypercohomology of Bloch’s higher Chow complex. Is there a quasi-isomorphism between this definition and the Cisinski-Deglise definition? Same question with rational coefficients, in which case we know the cohomology groups are the same, since they both agree with Adams eigenspaces of algebraic K-theory, but how does one get a quasi-isomorphism?
Matthias Wendt: is H_Deligne (in an unspecified sense) the unique spectrum representing Deligne coho? (akin to the unicity of BGL etc).
Malte Witte: can arithmetic K-theory be written as the K-groups of an appropriate Waldhausen category? (probably for this it would be sufficient to know the same question for Deligne coho, then a localization argument involving Waldhausen categories, which yields long exact sequences of K-theories, could apply).
Helene Esnault: does Burgos’ construction of the complexes representing Deligne cohomology also work with Q-coefficients, as opposed to R-coefficients, or (since I’m fairly sure it does not generalize easily): is there a way of having a spectrum representing H^Deligne(X, Q())?
Oliver Bräunling: is there a Bloch-Quillen-like spectral sequence expressing arithmetic K-theory as Zariski cohomology of an appropriate sheaf?
Two issues have been pointed out: I made the mistake of not making clear right away that HBhat is only defined after the choice of a lift of H_B –> H_D to the model category underlying SH, which causes the homotopy fiber to be defined up to non-unique isomorphism. This should be carved out more clearly in the paper. In particular: given two choices of the lift, the resulting Arakelov motivic cohomology groups don’t seem to be defined up to unique isomorphism (but up to non-unique), as far as I can see right now.
Secondly, but this just a question of presentation: Frédéric pointed out not all of the six Grothendieck functors are compatible with the inclusion DM(S) \subset SH(S)_Q. I don’t understand this, in particular not given their Corollary 12.3.3. I’ll have to discuss this with Frédéric more thoroughly. Also, I made the mistake of thinking that what is denoted DM_{BGL} in our paper (modules over the K-theory spectrum) is a full subcategory of SH. (The functor forgetting the H_B-module structure, though, i.e., DM_{Beilinson} \r SH, is fully faithful this is [CD, 13.2.8 + 5.3.37]). Both points just mean that we have to carefully state what category we currently work in–either DM_{BGL} for the purposes of arithmetic K-theory or DM_{Beilinson} for Arakelov mot. cohomology.
nLab page on Questions from others