Holmstrom Motivic cohomology over an arithmetic base

Remark: The equivalence between $H \Q$ and $H_{\Beilinson}$ (iso in $\DA (S, \Q)$ I think follows from the coniveau spectral sequence for the algebraic K-theory spectrum; it degenerates rationally.

Alternative to Voevodsky’s def of HZ: D-C in TCMM 10.2.1 define motivic cohomology, and say near the section on Voevodsky’s conjecture that the motivic cohomology spectrum with $\Lambda$-coefficients is $\gamma_* \mathbb{1}_S$, where $\gamma_* : \DMZ \to \DAoneZ$ is part of the premotivic adjunction of TCMM 10.4.1.

Cisinski in Paris: It should be true in general that HZ-modules identify with motives (note Roendigs et al have a restriction about $\SH(S)$ being generated by dualizable objects). The statement (for S = Spec(Z)?) should follow from absolute purity for Spec(Z), but can probably be proved without it. In general, there might be problems with good models for HZ over general base schemes. Later note: C and D are reworking the integral coefficients theory, and they feel free to change the def of motivic cohomology to get Voevodsky compatibility between different base schemes, as long as the definition stays the same for fields. Doing some change like this, it should be possible to get good compatible models, this might be the topic for the paper of D and C after the etale paper.

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