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I am looking for a reference, or a proof, of the following statement:

Let $f:Y\longrightarrow X$ be a smooth proper map of quasiprojective $K$ schemes, and let $\overline{f}:\overline{Y}\longrightarrow \overline{X}$ be the base change of $f$ with respect to $K\longrightarrow \overline{K}$ (assume characteristic zero).

Let $R^i\overline{f}\textbf{1}_Y$ denote the higher direct image of the trivial étale local system on $Y$, it is a local system on $X$. Proper base change tells us that its pullback under $\overline{X}\longrightarrow X$ is an étale local system on $\overline{X}$, which is identified with $R^i\overline{f}\textbf{1}_{\overline{Y}}$.

This gives the stalk of $R^i\overline{f}\textbf{1}_{\overline{Y}}$ at a geometric point $\overline{b}$ of $\overline{X}$, associated to a $K$-rational point $b$ of $X$, the structure of both a $\pi_1^{\text{et}}(\overline{X},\overline{b})$ and a $\pi_1^{\text{et}}(X,b)$-representation. The functoriality of the etale fundamental group induces a group homomorphism $\pi_1^{\text{et}}(\overline{X},\overline{b})\longrightarrow \pi_1^{\text{et}}(X,b)$, and the Grothendieck $\pi_1$-exact sequence tells us that this map is injective.

It seems reasonable to expect that the $\pi_1^{\text{et}}(\overline{X},\overline{b})$-representation, $(R^i\overline{f}\textbf{1}_{\overline{Y}})_{\overline{b}}$, is the restriction of the $\pi_1^{\text{et}}(X,b)$-representation $(R^i\overline{f}\textbf{1}_{\overline{Y}})_{\overline{b}}$. Is that indeed the case?

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    $\begingroup$ This is more or less the definition of the pullback from local systems on $X$ to local systems on $\bar X$, isn't it? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 26 at 17:06
  • $\begingroup$ Does this mean that $H^i_{et}(Y_b, K)$ and $H^i_{et}(\overline{Y}_{\overline{b}}, K)$ have the same dimensions? I find this confusing because in the case of a point, I think that some version of Hurewicz for etale cohomology should say that $H^1_{et}(\overline{K}, K) = 0$, while $H^1_{et}(K, K) = G_K^{ab}\otimes K$, am I wrong? $\endgroup$
    – kindasorta
    Commented Feb 26 at 18:32
  • $\begingroup$ Not sure what you mean by $\pi_1^{\operatorname{\acute et}}(X,b)$; that should be $\bar b$, right? Likewise, the only comparison between stalks is between the geometric stalks. There is no statement about $H^i(Y_b,K)$, only the representation of $\operatorname{Gal}(\bar K/\kappa(b))$ on $H^i(Y_{\bar b},K)$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 26 at 21:18

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