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. <https://mathoverflow.net/questions/89171/on-the-cohomology-of-g-m-bundles-and-purity-for-singular-varieties>



The Gysin long exact sequence for the complement of the zero section of a line bundle over a (possibly) singular base
Comment promoted to answer:


The following is Corollaire 1.5 in SGA5, Exposé VII:



> 
> Let $Z$ be any scheme, $p:X\to Z$ a rank $r$ vector bundle, $U=X-Z$, and $q:U\to Z$ the retsriction of $p$. Let $n$ be coprime to the residual characteristics of $Z$ and let $L$ be a sheaf of $\mathbb{Z}/n$-modules on $Z$. Then there is a long exact sequence
> 
> 
> $$\to H^{\ast-2r}(Z,L(-r))\to H^\ast(Z,L)\to H^\ast(U,q^\ast L)\to H^{\ast+1-2r}(Z,L(-r))\to$$
> 
> 
> 


That's how the corollary is stated but they also show that the sequence is naturally isomorphic to the more familar-looking sequence for cohomology with supports


$$\to H^{\ast}\_Z(X,p^\ast L)\to H^\ast(X,p^\ast L)\to H^\ast(U,q^\ast L)\to H^{\ast+1}\_Z(X,p^\ast L)\to$$


You should definitely have a look at the rest of exposé VII in SGA5, which proves a lot of related "expected" theorems for étale cohomology with very mild hypotheses (compared to what can be found elsewhere).


