I am reading etale cohomology and a question about Poincare duality comes to me. Suppose $k$ is a number field and $X/k$ is a smooth variety with dimension $n$ and $Y$ is a possibly singular variety also with dimension $n$, such that $X$ and $Y$ shares a common smooth open subset, i.e. there is a smooth variety $U$ with open embeddings \begin{equation} i:U \rightarrow X, ~~j:U \rightarrow Y \end{equation} and there exists a morphism $f:X \rightarrow Y$ that restricts to identity one U, i.e. \begin{equation} f \circ i=j \end{equation} For $\ell$ a prime number, there is an injective morphism between torison sheaves on $Y$, \begin{equation} j_!j^*(\mathbb{Z}/\ell \mathbb{Z}) \rightarrow \mathbb{Z}/\ell \mathbb{Z} \end{equation} from which we get a morphism from $H^n_{et,c}(\overline{U},\mathbb{Z}/\ell \mathbb{Z} ) $ to $ H^n_{et}(\overline{X},\mathbb{Z}/\ell \mathbb{Z} )$, \begin{equation} H_{et,c}^n(\overline{U},\mathbb{Z}/\ell \mathbb{Z} ) \rightarrow H^n_{et}(\overline{Y},\mathbb{Z}/\ell \mathbb{Z} ) \xrightarrow{f^*} H^n_{et}(\overline{X},\mathbb{Z}/\ell \mathbb{Z} ) \end{equation}
On the other hand $i$ induces a morphism \begin{equation} i^*:H^n_{et}(\overline{X},\mathbb{Z}/\ell \mathbb{Z} ) \rightarrow H^n_{et}(\overline{U},\mathbb{Z}/\ell \mathbb{Z} ) \end{equation} whose Poincare duality also determines a morphism from $H^n_{et,c}(\overline{U},\mathbb{Z}/\ell \mathbb{Z} ) $ to $ H^n_{et}(\overline{X},\mathbb{Z}/\ell \mathbb{Z} )$, \begin{equation} H^n_{et,c}(\overline{U},\mathbb{Z}/\ell \mathbb{Z} ) \rightarrow H^n_{et}(\overline{X},\mathbb{Z}/\ell \mathbb{Z} ) \end{equation}
Are the two morphisms from $H^n_{et,c}(\overline{U},\mathbb{Z}/\ell \mathbb{Z} ) $ to $ H^n_{et}(\overline{X},\mathbb{Z}/\ell \mathbb{Z} )$ the same?
More generally, are the two morphisms the same for other Weil cohomology theories when compactly supported cohomology are defined like the etale case?
