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?
Best Answer
This is because $U$ is again normal [Tag 033C], so it is irreducible since it is connected [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]. By irreducibility, there is only one generic point, so there is also only one point in the generic fibre.