Assume $X=\operatorname{Spec}(A)$ is connected and normal (especially integral), and let $g:\eta \hookrightarrow X$ be the inclusion of the generic point of $X$. In Milne's LEC script on Etale Cohomology (Example 12.4, page 81) is claimed that the stalk of derived direct image sheaf of sheaf $\mathcal{F}$ on etale site $(\eta)\_{et}=\operatorname{Spec}k(X))\_{et} $ in a geometric point $\overline{x}= \operatorname{Spec}k(x)^{\text{sep}} \to X$ is given by


$$ (R^rg\_\* \mathcal{F})\_{\overline{x}} =H^r(\operatorname{Spec}K\_{\overline{x}}, \widetilde{\mathcal{F}}) $$


where where $K\_{\overline{x}}$ is the field of fractions of $\mathcal{O}\_{X,\overline{x}}^{\text{et}}$, where the latter is the stalk of $\overline{x}$ in etale sheaf, ie the inductive limit of the system of connected etale affine neighborhoods $(U,u) \to X$ of $\overline{x}$.


$\widetilde{\mathcal{F}}$ is the pullback of $\mathcal{F}$ with respect the natural composition $\operatorname{Spec} \text{Frac}(\mathcal{O}\_{X,\overline{x}}^{\text{et}})
\to \operatorname{Spec}\mathcal{O}\_{X,\overline{x}}^{\text{et}} \to X$


**Question:** Why this identity holds? Can it be elaborated out in detail?


My considerations: In general we have for more general $g:Y \to X$ the stalk formula


$$ (R^rg\_\* \mathcal{F})\_{\overline{x}} = \varinjlim\_{(U,u) \text{ etale nbhd of } \overline{x}} H^r(U\_Y, U\_Y^\* \mathcal{F}) $$


where as before inductive limit is taken over the system of connected etale affine neighborhoods $(U,u) \to X$ of $\overline{x}$ and $U\_Y:= U \times\_X Y$.


Let's come back to $Y=\eta$ the generic point of $X$. By definition $\operatorname{Spec}\mathcal{O}\_{X,\overline{x}}^{\text{et}}=\varprojlim \_{\text{ etale nbhd of } \overline{x}}(U,u)= \operatorname{Spec} \varinjlim A\_U$


where $U=\operatorname{Spec}A\_U$. Moreover we have $(\varprojlim U) \times\_X \eta=\varprojlim (U\times\_X \eta)$


So once we know that $A\_U \otimes\_A k(X)= \text{Frac}(A\_U)$ we win, and it would be the exactly then the case if the generic fiber $\operatorname{Spec}A\_U \otimes\_A k(X)$ - which is etale over $\eta= \operatorname{Spec} k(X) $ by base change stability of etaleness- would be also *connected*. But why this should be the case here?


Does somebody see why the generic fiber $\operatorname{Spec}A\_U \otimes\_A k(X)$ at $\eta$ of an connected etale affine neighborhoods $(U,u)=\operatorname{Spec}A\_U \to X$ of $\overline{x}$ is connected, ie equals to a Spec of a field? Or at least that evenry such etale $A$-algebra is *dominated* by one whose generic fiber is connected.


If that's not the case, how else one should deduce that $ (R^rg\_\* \mathcal{F})\_{\overline{x}} =H^r(\operatorname{Spec}K\_{\overline{x}},\widetilde{\mathcal{F}} ) $ as claimed above in Milne's script?



Calculate stalk of etale derived pushforward sheaf (Milne's LEC)
This is because $U$ is again normal [Tag [033C](https://stacks.math.columbia.edu/tag/033C)], so it is irreducible since it is connected [Tag [033M](https://stacks.math.columbia.edu/tag/033M)]. Clearly the generic point of $U$ maps to the generic point of $X$, and conversely any point mapping to the generic point of $X$ is generic in $U$ since étale maps are (locally) quasi-finite [Tag [03WS](https://stacks.math.columbia.edu/tag/03WS)]. By irreducibility, there is only one generic point, so there is also only one point in the generic fibre.


