In Freitag and Kiehl's etale cohomology book p206,$H(X\_e,Rj\_\*(\Lambda|\_{X\_{\eta}}))\to H(X\_{es},i^\* Rj\_\*(\Lambda|\_{X\_{\eta}})$is a isomorphism,where X is a proper scheme and smooth over the base scheme S at all the point of Y, the complement of the scheme $X\_e$.


If $X\_e$ is proper,it is easily derived from the proper base change theorem.In this book,authors argue that,when F is locally constant,the $R\Gamma\_Y F$ has the base change property,but I don't understand how they do this(The purity theorem is only proved in the case that X and Y are both smooth.)So how to prove this formula?



How to prove this formula for nearby cycle?
In the case Freitag and Kiehl are describing ($X$ a family of quadrics degenerating to a node, $Y$ the divisor at $\infty$), $X$ and $Y$ are both smooth in a neighborhood of $Y$. This is sufficient for the base change property as it is a local condition.


