Poincare duality compatible with the definition of compactly supported cohomology in etale cohomology? I am reading etale cohomology and a question about the compatibility of Poincare duality and the definition of compactly supported cohomology comes to my mind.
Suppose $U$ is a smooth open sub-variety of a smooth variety $X$ of dimension $d$ defined over a number field $k$.
\begin{equation}
j:U \rightarrow X
\end{equation}
It induces a pull-back homomorphism,
\begin{equation}
j^*:H_{et}^{2d-r}(X,\mathbb{Z}/\ell \mathbb{Z}) \rightarrow H_{et}^{2d-r}(U,\mathbb{Z}/\ell \mathbb{Z})
\end{equation}
whose dual defines a homorphism
\begin{equation}
(j^*)^\vee:(H_{et}^{2d-r}(U,\mathbb{Z}/\ell \mathbb{Z}))^\vee(d) \rightarrow (H_{et}^{2d-r}(X,\mathbb{Z}/\ell \mathbb{Z}))^\vee(d)
\end{equation}
which by Poincare duality defines a homomorphism,
\begin{equation}
H_{et,c}^{r}(U,\mathbb{Z}/\ell \mathbb{Z}) \rightarrow H_{et,c}^{r}(X,\mathbb{Z}/\ell \mathbb{Z}) =H_{et}^{r}(X,\mathbb{Z}/\ell \mathbb{Z})
\end{equation}
On the other hand, from the definition of compactly supported cohomology group
\begin{equation}
H_{et,c}^{r}(U,\mathbb{Z}/\ell \mathbb{Z}):=H^r_{et}(X,j_!(\mathbb{Z}/\ell \mathbb{Z}))
\end{equation}
there is also a homorphism
\begin{equation}
H_{et,c}^{r}(U,\mathbb{Z}/\ell \mathbb{Z})=H^r_{et}(X,j_!j^*(\mathbb{Z}/\ell \mathbb{Z})) \rightarrow H^r_{et}(X,\mathbb{Z}/\ell \mathbb{Z})
\end{equation}
Are the two morphism from $H_{et,c}^{r}(U,\mathbb{Z}/\ell \mathbb{Z})$ to $H^r_{et}(X,\mathbb{Z}/\ell \mathbb{Z})$ the same?
 A: This is true. Let me work in the bounded derived category $D^b_{et}(X, \mathbb{Z}/\ell)$ first, and denote by $g: X \to Spec(k)$ the structure morphism. All morphisms in the following will be derived.
Then one can consider the counit map $$\phi: j_! j^* \mathbb{Z}/\ell \to \mathbb{Z}/\ell$$ and apply the duality operator $\mathbb{D}_X = \mathcal{RHom}(-, g^! \mathbb{Z}/\ell)$ to it. This gives 
$$\mathbb{D}_X(\phi):  \mathcal{RHom}(\mathbb{Z}/\ell, g^! \mathbb{Z}/\ell) \to \mathcal{RHom}(j_! j^* \mathbb{Z}/\ell, g^! \mathbb{Z}/\ell)$$
which, as $\mathcal{RHom}$ interacts favourably with the adjunctions, gets transformed into the unit map
$$\mathcal{RHom}(\mathbb{Z}/\ell, g^! \mathbb{Z}/\ell) \to j_*j^*\mathcal{RHom}(\mathbb{Z}/\ell, g^! \mathbb{Z}/\ell)$$
of the adjunction between $j_*$ and $j^*$. Finally, by Poincaré duality for $X$ smooth and proper (I assume you want $X$ to be proper!), we know $g^! \mathbb{Z}/\ell \cong \mathbb{Z}/\ell(d)[2d]$, so $\mathbb{D}_X(\phi)$ is just a shift of $$\psi: \mathbb{Z}/\ell \to j_* j^* \mathbb{Z}/\ell.$$
Now observe that $g_* j_! j^* \mathbb{Z}/\ell = R\Gamma_c(U, \mathbb{Z}/\ell)$ and $g_* j_* j^* \mathbb{Z}/\ell = R\Gamma(U, \mathbb{Z}/\ell)$, so $g_* \phi$ and a twist of $g_* \psi$ correspond exactly to your two maps of cohomology groups, which thus coincide. 
