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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?

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" 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.

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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. 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) or absolute Hodge cohomology (done in Brad Drew's thesis). The $\varphi$ from Déglise would be your realization and the $\mathrm{Td}_\varphi$ from Déglise would be 1 because the realization preserves Chern classes.

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