Holmstrom June 2010 notes on regulator comparison

Actual proof: There are at least 4 ways to approach this, one could give one proof now and more proofs in the next paper if appropriate. First of all one can pick any classical definition of the regulator one wants, for example via K-theory, Chern classas, the Bloch spectral sequence in Deninger and Scholl, or maybe something else in Nekovar, or Huber’s realizations paper (but then we need to check DM agrees in both settings and that everything is compatible).

One approach would then be via Gillet’s characteristic classes. My memory is that this is an argument on the level of cohomology groups so should be simpler than via Riou’s thesis. I think it’s the most realistic one for this week.

Another second approach is via Riou’s thesis. At some point, I think Lemma II.2.6 and Thm IV.11, he has some statements which are not so easy to understand (I don’t understand the notation completely), but the moral should be the following: Any definition of the regulator which can be lifted to the derived category (of complexes, spectra, or whatever), must agree with the one we define, by some uniqueness statement. So one needs a classical definition of the regulator which has such a lift, I imagine this should certainly be possible. And if not, maybe you can show by some trick or modification or abstract argument that a classical definition can be lifted. Actually you probably need more than the above theorems in themselves, you need some more of his underlying theory and you need to check that what he does there for motivic cohomology also works for Deligne cohomology, or something like this. He said this is perfectly fine but not written down in the thesis. We could always email him and ask things if necessary. See notes from long discussion with Riou about this approach.

The third approach is also via Riou’s thesis. According to Cisinski, there is somewhere some kind of compatibility statement between the system of Chern classes. Now this should mean that if $c_1$ agree, then if both systems of Chern classes (i.e. our regulator and the classical regulator) agree. This uses some kind of good properties/compatibility (with twists etc involved) which we must verify for both sides. We must also find where this is stated in Riou’s thesis. The point of this approach would be that we don’t need a lift of the classical regulator to a derived category. Maybe a more precise formulation is the following, from my notes: “To prove agreement without a map in the derived category (the plus version?) it could be enough to show that the classical regulator or Chern classes are ”compatible with twists“ see Riou’s thesis or his arXiv paper. Then a lift does exist, to $\SH$ say, and then uniqueness applies.”

A fourth approach is by tracking through our construction somehow and using that we put in the first Chern class to Deligne cohomology at a certain point, and hence should get out the same thing, and that the first Chern class determines all Chern classes/the whole regulator, by some compatibility statement and/or the splitting principle.

Am not sure if Feliu’s approach would be of any help: She uses Grassmannians and BGL I think, maybe we can mimick this somehow. Or obtain agreement with the Beilinson regulator via comparison maps relating our construction with Feliu’s groups.

See Feliu, end of 1.4.6, for maps $c_1: Pic(X) \to H^2(X, \R(1))$ and $c_1: \Gamma(X, \Gm) \to H^1(X, \R(1))$ (both Deligne cohom).

Jakob mentioned a description of the map from Pic at the end of Esnault-Viehweg.

We should mention other places were similar problems have been encountered, for example the Landweber paper. It could also be that this paper helps us prove what we want.

See also the recent Documenta paper of Burgos et al. This might give a new way of proving agreement.

A very very naive and probably useless argument using agreement of kernels, see pages R1-R2.

Maybe it is useful to have the description of the thing which represents Pic in the effective category: This is the object $\Gm \otimes_{\Z}^L R[1]$ (see MWC 1.1.19).

See MWC 2.3.23-24 for a theorem about Chern classes in the setting of mixed Weil cohomologies, possibly it contains something useful, in particular see the characterization of cycle class maps in terms of functoriality and compatibility with cup product and first Chern classes. See also the references to Riou given here.

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Other comparison theorems notes:

Idea here (was this really for comparison theorems??): start with a sheafication of Bloch’s higher Chow groups as Suslin-Friedlander (???), then apply some kind of Cisinski argument based on initial object in a monoid cat. Another idea: obtain actual comparison results between the pullback of our spectrum and Feliu and Goncharov by passing via comparison between the Beilinson spectrum and Voevodsky’s spectrum, and between Voevodsky’s spectrum and ordinary Chow groups. Note that the latter might be possible to do with almost integral coefficients only using the project of Kelly.

Compare with some flavour of arithmetic K-theory? What about redefining arithmetic K-theory with our methods?

We should also compare our definition of the height pairing with the classical one, this deserves a separate heading if we can achieve this.

Discuss comparison between Chow groups and motivic cohomology over an arithmetic base: what is known and what is expected?

nLab page on June 2010 notes on regulator comparison