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Let $f:X\rightarrow Y$ be a proper birational morphism of smooth projective varieties over $k=\bar k$ ($char(k)>0$). Is it true that $H^1(Y,\mathbb Z_l)\stackrel{f^*}\simeq H^1(X,\mathbb Z_l)$ ($l\neq char(k)$) ?

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Yes. It's sufficient to prove an isomorphism on mod $\ell^n$ cohomology for all $n$.

But using the exact sequence $\mu_{\ell^n} \to \mathbb G_m \to \mathbb G_m$, it is sufficient to prove that $H^1(X, \mathbb G_m) \to H^1(Y, \mathbb G_m)$ is an isomorphism on the identity component.

So it follows from birational invariance of the Picard variety which I think you can prove directly.

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  • $\begingroup$ Thank you very much for you help. Could you, please, giveme a reference proving this birational invariance of $Pic^0$ (or $Alb$) ? $\endgroup$ – user100915 Sep 12 '15 at 20:26
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Lang's Abelian Varieties asserts the birational invariance of the Albanese is trivial. For some pointer of how to do it, say $U$ is an open subvariety of $X$ and $f: U \to Y$ is dominant and is an isomorphism onto $f(U)$. Then $U$ has a canonical embedding into both $Alb(X)$ and $Alb(Y)$, and then you can get a map from $X$ to $Alb(Y)$. This should give you what you want.

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