Let $R$ be a henselian dvr, $s,\eta\in\text{Spec}(R)$ the closed and generic points, and $f : X\to \text{Spec}(R)$ a proper smooth scheme.


For a prime $\ell$ invertible on $R$, is there a specialization map


$$sp^i\_{\eta,s} : R^if\_\*(\mu\_{\ell^n})\_{s}\to R^if\_\*(\mu\_{\ell^n})\_{\eta}$$


(with $\eta$ and $s$, not the geometric points over them)?


Is it an isomorphism, injective, surjective?



Specialization map étale cohomology
If $\ell$ is invertible in $R$, then $R^i f\_\* (\mu\_{\ell^n}) $ is a locally constant sheaf on $R$ by smooth and proper base change. Hence it is a represenation of the fundamental group of $R$, which is equal to the Galois group of the residue field, because $R$ is Henselian.


This fundamental group is easily seen to be a quotient of the Galois groups of both $s$ and $\eta$, and hence the "stalks" at $s$ and $\eta$, which are the Galois-invariants of the geometric stalks, are both equal to the $\pi\_1$-invariants, hence naturally isomorphic.


