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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.

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1 Answer 1

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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.

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