# How to prove this formula for nearby cycle?

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?

• Can you add more details and explain all the notation? What is $X_{es}$? Jun 3, 2017 at 13:10
• 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 Jun 3, 2017 at 13:28

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.