See file Arakelov-mot.tex.
Ref to Deninger-Nart somewhere?
Sort out the ring structure on Deligne cohom, via weak Hodge complexes, operads, or otherwise.
See all three letters of Cisinski on the subject, in pdf format somewhere.
Jakob email Oct 2010: you mentioned a couple of times that you study or studied the ring structure of our Arakelov spectrum. I just came across this: let \hat 1 be the Arakelov spectrum, 1 (as usual) the Beilinson spectrum. Using the 1-module structure of \hat 1, we can simply define
\hat 1 \tensor \hat 1 \rightarrow (identity \tensor forgetful map) \hat 1 \tensor 1 \rightarrow (module structure) \hat 1
Is this the ring structure you envisioned?
Deglise email:
Actually, you can take a look at the book of Levine, chapter 5. It contains a suitable definition of a Bloch-Ogus theory and show of to construct a realisation theorem (in his setting).
Is it reasonable to expect anything roughly along the following lines, maybe with some additional hypotheses: “any cohomology which is homotopy invariant and expressed as abelian sheaf hypercohomology of a sheaf of commutative DGAs is represented by a strict ring spectrum”? Are there precise such statements, or is the problem more difficult than this?
Concerning the first statement: take a sheaf over Sm/S of commutative DGA E such that its Nisnevich hypercohomology H is homotopy invariant, the object E gives an object of the derived category of Nisnevich sheaves, which is by definition A^1-local, thus this object represents the hypercohomology E in D_{A^1}^{eff}(S) and it has a strict ring structures (by definition). The problem is to get an object in the non effective category which still represents H. For this, you have at least to assume: (stability) H^(X \otimes T,T \otimes E)=H^(X,E), for any smooth X/S where T=Z(G_m/1) is the Tate object. This which will give you a map T->T \otimes E in the derived category (as the image of 1 by the preceding isomorphism for example). You have to use it to construct a symmetric T-spectrum - see for example the construction of the P^1-spectrum representing homotopy invariant K-theory (in the simplicial setting) in the preprint of Riou: arXiv:0907.2710. This spectrum will represents H by construction. The problem is to preserve the strict ring structure in the process (this is a hard problem in the case of K-theory for example). With Denis-Charles, we have done such a general construction in the case of mixed Weil cohomology (arXiv:0712.3291, construction in 2.1.5), but it uses a stronger assumption than the above one (which is not satisfied by Deligne cohomology !) May be if you get a good map: T->T \otimes E when E is Deligne cohomology, you can use the same strategy. (Note also that working with rational coefficients should make the situation easier)
nLab page on Peter project