Suppose $X$ is a variety defined over an algebraically closed field $k$ of characteristic zero, and $Z$ is a closed subvariety of codimension $c$, then there is a long exact sequence of etale cohomology groups, \begin{equation} \cdots \rightarrow H^r_{Z}(X,\Lambda) \rightarrow H^r(X,\Lambda) \rightarrow H^r(U,\Lambda) \rightarrow H^{r+1}_Z(X,\Lambda) \rightarrow \cdots \end{equation} where $U:=X-Z$ and $\Lambda:=\mathbb{Z}/\ell \mathbb{Z}$. What's more, the group $H^r_Z(X,\Lambda)$ vanishes when $r <2c$, so we have \begin{equation} 0 \rightarrow H^{2c-1}(X, \Lambda)(c) \rightarrow H^{2c-1}(U,\Lambda)(c) \rightarrow H^{2c}_Z(X,\Lambda)(c) \xrightarrow{f} H^{2c}(X,\Lambda)(c)\rightarrow \cdots \end{equation} From the paper Beilinson's Conjectures, there is a cycle class map \begin{equation} cl(Z):\Lambda(0) \rightarrow H^{2c}_Z(X,\Lambda)(c) \end{equation} such that the composition with $f$ \begin{equation} f \circ cl(Z):\Lambda(0) \rightarrow H^{2c}(X,\Lambda)(c) \end{equation} is the usual cycle class map.
Question 1: could anyone explain how to define the cycle class map $cl(Z)$, and show its composition with $f$ is the usual cycle class map?
When both $X$ and $Z$ are smooth, then $cl(Z)$ is just the Gysin isomorphism.
Question 2: Do the above constructions work when $Z$ is only a closed subscheme, e.g. $Z$ is not reduced?