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I have an argument, which I wonder if someone could check:

Let $X$ be an irreducible reduced scheme over a field $k$. Then we have a normal scheme $X^{norm}$ with a finite birational $f:X^{norm}\rightarrow X$. Then for any étale sheaf $\mathcal{F}$, the higher direct image $R^if_*\mathcal{F}$ vanishes for $i>0$. In particular, the Lerray spectral sequence yields $$H^p_{ét}(X,f_*\mathcal{F})=H^p_{ét}(X^{norm},\mathcal{F}).$$ However, I would like understand $H^i_{ét}(X,\mathbb{Z}/n)$ and relate it to the cohomology of $X^{norm}$, and unfortunatley $f_*(\mathbb{Z}/n)\neq \mathbb{Z}/n$. Hence my question is

Is there a good way to relate the $\ell$-adic cohomology of a scheme with the $\ell$-adic cohomology of its normalization?

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    $\begingroup$ I'm afraid that this result is not true (it fails for e.g. a nodal curve. One way to see this is to compute the singular cohomology over $\mathbb{C}$.) The issue is that the math.SE link you give is only for the Zariski topology, not the etale topology (see the comment of Roland.) $\endgroup$
    – dhy
    Commented Jul 25, 2020 at 23:56
  • $\begingroup$ Ah, good, that makes sense, thank you. Is there a way to relate the $\ell$-adic cohomology of the normalization of a curve with the $\ell$-adic cohomology of the curve itself? $\endgroup$
    – curious math guy
    Commented Jul 26, 2020 at 0:49
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    $\begingroup$ The best way would be to understand the cokernel of the map. $\mathbb Z/n \to f_* \mathbb Z/n$. This is supported in the closed subset of $X$ where the fiber of $f$ is disconnected. So a first step would be to identify that closed subset. $\endgroup$
    – Will Sawin
    Commented Jul 26, 2020 at 1:09
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    $\begingroup$ If I remember correctly, there is actually an equivalence for étale cohomology between X and its absolute weak normalization. That's because the map $X^{awn}\to X$ induces an equivalence of étale ∞-toposes. Over a field of characteristic $0$, this is the seminormalization, and over a field of characteristic $p>0$, it is the perfection. I think in general this is the best you can do. $\endgroup$
    – Harry Gindi
    Commented Jul 26, 2020 at 1:46
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    $\begingroup$ Why say "$\infty$-topos" when you can just say "site"? $\endgroup$
    – David Hansen
    Commented Jul 27, 2020 at 12:16

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