I have a question about pushforward maps for the motivic cohomology groups $H^p(X, \mathbf{Q}(q))$, for $X$ a smooth variety over a characteristic 0 field.


According to Mazza--Voevodsky--Weibel "Lectures on motivic cohomology", these groups have the following *covariant* functoriality properties (in addition to their usual contravariant functoriality)


1. if $f: X \to Y$ is a proper flat morphism, with fibres of dimension $d$, there is a pushforward map $f\_\*: H^{p + 2d}(X, \mathbf{Q}(q + d)) \to H^{p}(Y, \mathbf{Q}(q))$;
2. if $f: X \to Y$ is the inclusion of a closed subvariety of codimension $c$, then there is a pushforward map $f\_\*: H^{p}(X, \mathbf{Q}(q)) \to H^{p + 2c}(Y, \mathbf{Q}(q + c))$.


(Note that the first map lowers the cohomological degree, while the second map raises it.)


There are also corresponding maps in $\ell$-adic étale cohomology (for any prime $\ell$), or in Beilinson's absolute Hodge cohomology when the base field is $\mathbf{R}$ or $\mathbf{C}$; and there are "realisation" maps from motivic cohomology to these other two cohomology theories.



> 
> Are the pushforward maps (1) and (2) known to be compatible with their étale and Hodge analogues under the realisation maps?
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> 
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I guess this must be known, but I've been unable to find a reference which states this directly. For étale cohomology, the work of Ciskinski--Deglise ["Etale motives"](http://arxiv.org/abs/1305.5361) apparently gives a realisation functor (on some category of motivic complexes) that is "compatible with Grothendieck's six operations", which sounds promising, but I'm not sufficiently expert to know if the concrete compatibility statements I want follow from this. And I've been unable to find any reference treating the case of absolute Hodge cohomology.



Motivic cohomology and pushforward maps
The answer is **yes, they are compatible**. This is consequence of the machinery developped for the Riemann-Roch theorem. 


Recall that motivic cohomology, $l$-adic cohomology, and absolute Hodge cohomology have *additive* Chern classes (that is to say, $c\_1(L\otimes L')=c\_1(L)+c\_1(L')$) and that the realization maps Chern classes to Chern classes. You should take a look to [Déglise's preprint](http://arxiv.org/pdf/1111.4203v2.pdf). The theorem you use is 4.3.2. The spectra $\mathbb{E}=\mathrm{H}\_B$, Beilinson's motivic cohomology spectrum, and $\mathbb{F}$ the spectrum representing either $l$-adic cohomology (done [here](http://arxiv.org/pdf/0712.3291.pdf)) or absolute Hodge cohomology (done in Brad Drew's [thesis](http://perso.ens-lyon.fr/frederic.deglise/docs/2015/these_drew.pdf)). The $\varphi$ from Déglise would be your realization and the $\mathrm{Td}\_\varphi$ from Déglise would be 1 because the realization preserves Chern classes.


