For a smooth projective variety $X$ over any field $K$, Voevodsky showed in his paper ``Motivic Cohomology Groups Are Isomorphic to Higher Chow Groups in Any Characteristic" that the motivic cohomology group $H^{2j}_{\mathcal{M}}(X, \mathbb{Z}(j))$ is isomorphic to $CH^j(X)$, the Chow group of codimension $j$ cycles modulo rational equivalence.

There are regulator maps to etale cohomology ($l \neq char \ K$) $$ H^{2j}_{\mathcal{M}}(X, \mathbb{Z}(j)) \to H^{2j}_{et}(X_{\bar{K}}, \mathbb{Z}_{l}(j)).$$

There is also a cycle class map $$ CH^j(X) \to H^{2j}_{et}(X_{\bar{K}}, \mathbb{Z}_l(j)).$$


Where is it shown that the regulator maps agree with the cycle class maps, under Voevodsky's isomorphism?



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