Let $X$ be a smooth variety over an algebraically closed field (whose characteristic could be positive), $Y\to X$ is a $G\_m$-bundle ($G\_m=\mathbb{A}\setminus \{0\}$). Then I want to have a long exact sequence that relates the \'etale cohomology of $Y$ with the one of $X$; this should probably be similar to Theorem 3.57 of <http://books.google.ru/books?id=wJJ0fnq0LPMC&pg=PA138&lpg=PA138&dq=long+exact+sequence+circle+bundle+cohomology&source=bl&ots=-brkx3H2yi&sig=HmC1xKeRfXqB4G5CE53uG5PrjtU&hl=ru&sa=X&ei=8N0ZT5PrPMXsOZ-8sJ8L&ved=0CGMQ6AEwCA#v=onepage&q=long%20exact%20sequence%20circle%20bundle%20cohomology&f=false>
Is such a fact known/true?



The cohomology of a $G_m$-bundle
This is true, if you take étale cohomology with coefficients in a finite abelian group of order not divisible by the characteristic. You can embed $Y$ in the corresponding line bundle $L \to X$. Then by smooth base change the pullback from the cohomology of $X$ to that of $L$ is an isomorphism, and you can apply the Gysin sequence of $Y$ in $L$ (see for example <http://www.jmilne.org/math/CourseNotes/LEC.pdf>, Chapter 16).


