Jannsen's generalized Tate conjecture for arbitrary varieties

Let $k$ be a finitely generated field and $X$ a variety over $k$. For a prime $\ell\neq\operatorname{char}(k)$, let $H_{2i}(\overline{X},\mathbb{Q}_{\ell}(i))$ denote $\ell$-adic étale homology and let $G_k$ be the absolute Galois group of $k$. The cycle class map is

$$\operatorname{CH}_i(X)\otimes_{\mathbb{Q}}\mathbb{Q}_{\ell}\to H_{2i}(\overline{X},\mathbb{Q}_{\ell}(i))^{G_k}.$$

Jannsen's generalized Tate conjecture. This cycle class map is surjective. It extends the smooth-variety formulation of the Tate conjecture to arbitrary varieties using étale homology.