Comparison of cycle maps

Question

Let $X$ be an algebraic variety over $\bar{\mathbb{Q}}$ of dimension $d$, then there is the l-adic cycle map $\mathrm{cl}_{et}:\mathrm{CH}^i(X)\rightarrow\mathrm{H}^{2i}(X,\mathbb{Q}_\ell(i))$ from the Chow group into the l-adic etale cohomology group, where $\mathbb{Q}_\ell(i)$ refers to the Tate twist. Its definition is as follows: given an irreducible subvariety $Z$ of codimension $i$, there is a pullback map $j^\ast:\mathrm{H}^{2(d-i)}_c(X,\mathrm{Z}/n\mathrm{Z})\rightarrow H^{2(d-i)}_c(Z,\mathrm{Z}/n\mathrm{Z})$; based on the Poincare duality $\mathrm{H}^{2(d-i)}_c\big(X,(\mathrm{Z}/n\mathrm{Z})(d-i)\big)\times \mathrm{H}^{2i}\big(X,(\mathrm{Z}/n\mathrm{Z})(i)\big)\rightarrow \mathrm{H}^{2d}_c(X,(\mathrm{Z}/n\mathrm{Z})(d))\cong \mathrm{Z}/n\mathrm{Z}$, one defines $\mathrm{cl}_{et}(Z)$ to be the class in $\mathrm{H}^{2i}\big(X,\mathrm{Z}/n\mathrm{Z}(i)\big)$ representing the homomorphism $j^\ast(d-i)$ on $\mathrm{H}^{2(d-i)}_c\big(X,\mathrm{Z}/n\mathrm{Z}(d-i)\big)$. By passing to the inverse limit, one arrives at the coefficient sheaf of $\mathbb{Z}_\ell$ and $\mathbb{Q}_\ell$.

When $X$ is nonsingular, there is also the cycle map $\mathrm{cl}_\mathrm{C}:\mathrm{CH}^i(X)\rightarrow\mathrm{H}^{2i}(X(\mathbb{C}),\mathrm{C})$ into the Singular cohomology group. So what is the relation between these two cycle maps? (Is there a reference on this question?) If one embeds $\mathbb{Q}_\ell$ into its topological algebraic closure $\mathbb{C}_\ell\cong \mathbb{C}$, would $\mathrm{cl}_{et}$ and $\mathrm{cl}_\mathrm{C}$ coincide or, if not, how to compare the $\mathrm{Z}$-module structure of the two images?

In Milne's lecture notes on Etale cohomology, he states a comparison theorem (Thm. 21.1) between etale cohomology and the singular cohomology based on the complex topology. According to it, there is a canonical isomorphism $\mathrm{H}^{2i}_{et}(X,\mathrm{Z}/n\mathrm{Z})\cong \mathrm{H}^{2i}(X(\mathbb{C}),\mathrm{Z}/n\mathrm{Z})$. It is with respect to such an isomorphism that one asks about the compatibility of the cycle maps.

Answer 1

Everything you could wish for is true :-). Passing to the inverse limit in Milne's theorem 21.1 one gets an isomorphism

$$ H^i_{et}(X, \mathbf{Z}_\ell) = H^i_{sing}(X(\mathbf{C}), \mathbf{Z}) \otimes \mathbf{Z}_{\ell}$$

(you don't have to extend scalars to $\mathbf{C}$ or $\mathbf{C}_\ell$ or anything nasty like that). Moreover, the same works if you put $H^i_c$ instead of $H^i$ on both sides, and the isomorphism is compatible with cycle classes. The way to see that is as follows: it's obvious if the cycle is the whole of $X$; and the comparison map is functorial in $X$ and respects Poincare duality, so it follows for an arbitrary cycle. (The same compatibility also works for Chern classes on higher $K$-theory, with the cycle-class case being $K_0$.)