Let $pr:X\to Z$ be a line bundle of (possibly) singular varieties (that could be reducible) over a characteristic $p$ field ($p$ could be zero); $U$ is the complement to the zero section $Z\to X$. Then for any $l\neq p$, $n>0$, there should exist a Gysin long exact sequence for the (etale or singular) cohomology $\dots \to H^i(X,\mathbb{Z}/l^n\mathbb{Z}) \to H^i(U,\mathbb{Z}/l^n\mathbb{Z})\to H^{i-1}(Z,\mathbb{Z}/l^n\mathbb{Z}(-1))\to \dots$ that is functorial in $pr$. Is this true?

Note that one can compute the cohomology of $U$ as the hypercohomology $H^*(Z,Rpr'_\ast\mathbb{Z}/l^n\mathbb{Z}_U)$, where $pr': U\to Z$ is the corresponding prinicple $G_m$-bundle. Hence the problem is to verify that $R^1pr'_*\mathbb{Z}/l^n\mathbb{Z}_U\cong \mathbb{Z}/l^n\mathbb{Z}_Z(1)$. Certainly, there is such an isomorphism for the trivial $G_m$-bundle $U\cong G_m\times Z$. Since this isomorphism does not seem to depend on the choice of a trivialization, one can 'glue' 'global' functorial isomorphisms from these 'local' isomorphisms. Yet I would prefer to have a reference for this result (or for some similar one).

Cf. On the cohomology of G_m-bundles and purity for singular varieties