For the three classical defs of ht pairing, see refs in Schneider’s survey. Maybe I can compare Bloch’s pairing with our pairing?
Check Feliu’s uniqueness paper.
Jakob email Aug 2010: I think I have an argument proving that the universal HB-module structure does induce the Beilinson regulator. I’ll try to send you a version tomorrow, but the basic idea is simple, essentially the way I told you in Barcelona: using the Burgos/Wang work, we get a map from the S-construction to our complex calculating Del. cohomology. This is the Beilinson regulator. Then, we cook up a map of spectra out of this. On the left, we have this BGL spectrum, as Riou denotes it. On the right we have the Deligne spectrum. Here, and I’m quite happy I now clearly understand this, the commutativity of the regulator maps at the individual levels of the spectrum with the bonding maps of both spectra requires do take exactly the definition of the spectrum you once proposed. Nice, isn’t it? OK, so we have a map of spectra and using Riou’s work, we lift it up to a map in SH. Now, we have a map (of ring spectra, in SH) HB \rightarrow D. We use it do endow D with a HB-model structure in the usual way. We thus get an HB-algebra structure on D. But, and this is the key point, once we know (what we do know) that D is orientable, there is a unique HB-algebra structure. In other words, the universal one and the one cooked up manually must agree.
I previously tried to use the work of Gillet instead of the Burgos/Wang paper. However, using this is more complicated, since the definition of Chern character (i.e., Beilinson regulator in our case) K_i(X) \r H^{2p-i}_Deligne (X, p) depends on i (for i=0 there is sort of an exception in the formulae). Therefore, this is less well-suited for general arguments like this.
Jakob email, August 2010: A piece of good news: I proved the agreement of our groups (X projective and regular, which is the generality Takeda, Feliu and GS have) with arithmetic K-theory. “Essentially, this is an elongated exercise in handling the various model category structures used to define . ” See section 3.4. Thus, by GS’s work, our groups also agree with their arithmetic Chow groups. The simple idea is this: lift the homotopy fiber of a map of simplicial sets (namely the one coming from the S-construction of K-theory to Burgos’ Deligne coho. complex) to SH. In doing this, you have to show that the objects you take the homotopy fiber of are fibrant (in SH). The least trivial property going in here (IMO) is the Nisnevich locality of the Waldhausen S-construction. I was quite happy to understand this. Of course, the format is still rough, partly because I stopped polishing it for now, since we have to agree on notation (the sooner the better). I’m somehow getting into the mood of working in SH as long as possible. Another minor notational thing: we need to have a spectrum which contains Deligne cohomology in all twists (so that it can receive a map from K-theory) and a subspectrum having only one particular twist (so that it receives a map from motivic cohomology).
nLab page on Comparison theorems