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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?

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  • $\begingroup$ Can you add more details and explain all the notation? What is $X_{es}$? $\endgroup$ – Will Sawin Jun 3 '17 at 13:10
  • $\begingroup$ Oh,I'm sorry.s is the close point of S, Xes is the fiber.\eta is the algebraic geometry point,the closure of the genertic point,j is the immerse of the fiber of \eta $\endgroup$ – wongdl Jun 3 '17 at 13:28
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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.

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