Question

I would like to relate the (etale or singular) cohomology of a $G_m$-bundle $B$ over a variety $X$ over an algebraically closed field with the one of $X$ ($X$ is a local complete intersection). Now, we can present $B$ as the complement to $X$ of an $\mathbb{A}^1$-bundle $B'/X$; since (etale) cohomology is homotopy invariant, we have $H^\ast(B')\cong H^*(X)$. Now, similarly to the case when $X$ is smooth (see the answer to http://mathoverflow.net/questions/86688/the-cohomology-of-a-g-m-bundle) I would like to write the Gysin long exact sequence. I suspect that I can do this in my case also, since '$X$ is a nice subvariety of $B'$' (I should look at its normal bundle here?). Is this true? A related question: for a closed embedding $i$, when the Absolute Purity isomorphism is fulfilled?