If $k$ is a perfect field (not necessarily separably closed), $f : X\to Y$ a proper birational map of smooth projective $k$-varieties, $\ell$ a prime invertible in $k$, is the induced map, for **some** $j$


$$f^\* : H^j\_{ét}(Y, \mu\_{\ell^n}^{\otimes i})\to H^j\_{ét}(X, \mu\_{\ell^n}^{\otimes i})$$
an isomorphism, for every $n\ge 1$?


For example, if $k$ is of characteristic zero and $i=1$, $j=2$, we have


$$0\to \text{CH}^1(X)/\ell^n\to H^2(X,\mu\_{\ell^n})\to \text{Br}(X)[\ell^n]\to 0$$


and left and right sides are birational invariants.



Birational invariants in étale cohomology
The answer is no.


Let $Y$ be a smooth projective surface over $k$, $f : X\to Y$ the blowup of a point. Then $f^\* : \text{CH}^1(Y)\to\text{CH}^1(X)$ is *not* an isomorphism, and for some $n\ge 1$ neither is its reduction mod $\ell^n$.


Your exact sequence shows $i=1, j=2$ and the above choice of $X,Y,f$ is a counterexample.


