Consider the (complex) geometric realization of the motivic cohomology theory on simplicial presheaves over complex smooth schemes, which is a functorial homomorphism of $R$-modules, where $R$ is the coefficient ring:
\begin{equation}\label{eq}
 \varphi: H^{s,t}\_{mot}(X;R)\to H^s(X(\mathbb{C});R).
\end{equation}


When restricted to the Chow ring $\varphi:H^{2\*,\*}\_{mot}(X;\mathbb{Z})\to H^{2\*}(X(\mathbb{C});\mathbb{Z})$ and $X$ a complex algebraic variety, it is a classical result that $\varphi$ is the cycle class map ring homomorphism.


Generally, is the map $\varphi: H^{\*,\*}\_{mot}(X;R)\to H^\*(X(\mathbb{C});R)$ an R-algebra homomorphism? I was not able to find a direct proof of this result in the literature. Or perhaps this is an obvious fact that I am missing.



The multiplicativity of the (complex) geometric realization of motivic cohomology
Complex realization sends the motivic Eilenberg-MacLane spectrum to the classical Eilenberg-MacLane spectrum [Theorem 5.5 in Marc Levine's "A comparison of motivic and classical stable homotopy theories]. It is also a symmetric monoidal functor, hence preserves ring spectra. The result follows.


