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Let $f:X\to Y$ be a proper morphism, and $\mathcal{F}$ be a coherent sheaf on the small \'{e}tale site $X_{et}$, flat over $Y$. For each geometric point $\bar{y}\to Y$, regard the pull back $\mathcal{F}_{\bar{y}}$ as a coherent sheaf for the Zarisky topology on $X_{\bar{y}}$. Define the following function $h^i(\bar{y},\mathcal{F})=\dim_{\Omega(\bar{y})}H^i(X_{\bar{y}},\mathcal{F}_{\bar{y}})$. I want to know whether the theorems Chap III 12.8, 12.9, 12.11 in Hartshorne are still true for this function $h^i$ on all geometric points? I appreciate it if anyone can give a reference if the statements are true.

Maybe I need to explain my concern by distinguishing a geometric point and a point. Let $\bar{y}\to Y$ be a geometric point of $Y$ with the image $y\in Y$. Let $k(y)$ denote the residue field of the point $y$, and $\Omega(\bar{y})$ denote the separably algebraically closed field containing $k(y)$. I think the statements would follow from the following statement $\dim_{\Omega(\bar{y})}H^i(X_{\bar{y}},\mathcal{F}_{\bar{y}})=\dim_{k(y)}H^i(X_y,\mathcal{F}_y)$. But this is not obvious to me. Am I being stupid?

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Quasi-coherent sheaves on the small étale site of a scheme are the same ordinary quasi-coherent sheaves, by descent (see stacks.math.columbia.edu/tag/03DR ). At least if your schemes are locally noetherian, the same should be true for coherent sheaves, and flatness over $Y$ should be preserved as well, so I don't see what could go wrong. –  Mattia Talpo Mar 15 at 11:51
    
Yes, I think you are right, the sheaf is the ordinary coherent sheaf flat over $Y$. What I am worried is the that the function $h^i$ is defined for all geometric points of $Y$ not points of $Y$. This is different from the statements in Hartshorne. –  user38276 Mar 16 at 16:08
    
$H^i(X_{\bar{y}},\mathcal{F}_{\bar{y}})$ is canonically isomorphic to $H^i(X_y,\mathcal{F}_y)\otimes _{k(y)}\Omega (\bar{y})$, see Hartshorne, Proposition III.9.3. –  abx Mar 16 at 17:20

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