I have question about a statement from *Etale Cohomology and the Weil Conjecture* by Freitag, Kiehl
at the top of page 16. It seemingly uses the same notations as introduced at the bottom of page 15
and is seemingly a consequence of following two facts.


Let $A$ be a *strict Henselian* ring (i.e. Henselian + residue class field separably algebraically closed).  

(This book works with definition 1.17, p 15, a noetherian local rings $A$ is *Henselian* if
every local quasi-finite homomorphism from $A$ to a localization of a finitely generated
$A$-algebra is *finite*).  

Next, the authors make a remark and deduce a consequence:


\*1.18 Remark.\*Let A be a strictly Henselian ring. Then every local-etale homomorphism
$A \to B$ is an isomorphism.  

As a consequence it can be deduced from this that For every faithfully flat etale homomorphism
$\varphi: A \to B$ of a strictly Henselian ring $A$ into a finitely generated $A$-algebra $B$ there exist
a right inverse; that is a map $\psi:B \to A$ with $\psi \circ \varphi= id\_A$ ( [here an expanation](https://mathoverflow.net/questions/400750/faithfully-flat-etale-morphism-from-strictly-henselian-ring-from-etale-cohomolo)
why this consequence is true)


Now comes a claim I not understand. Next is said that for *any* sheaf of abelian groups $\mathcal{F}$ the
functor


$$ \operatorname{Et}(B) \to \operatorname{(Ab)}, \ C \mapsto \mathcal{F}(C) $$


is exact. Why that's true? (here $\operatorname{Et}(B)$ is the category of etale extensions of $B$
understood as full subcategory of the category of $A$-algebras)


*Question 1:*  

My first confusion is that originally in the book the top line defining the functor reads as


$$ \operatorname{Et}(B) \to \operatorname{(Ab)}, \ B \mapsto \mathcal{F}(B) $$


This not make any sense, so obviously the author meant something else. What?  

I guess (but not know) that the authors maybe intended to write $A$ instead of $B$, i.e.maybe to work with $\operatorname{Et}(A)$ etale extensions of a strict Henselian base; but I don't know, that's only a guess of mine, which "looks" for me more "natural". Does anybody see what the authors had there in mind?


*Question 2:* Why the statement about the exactness of the functor above is true?  

We have to show that for every exact sequence


$$ 0 \to R\_1 \to R\_2 \to R\_3 \to 0 $$


of etale $B$-algebras die Sequenz of abelain groups


$$ 0 \to \mathcal{F}(R\_1) \to \mathcal{F}(R\_2) \to \mathcal{F}(R\_3) \to 0 $$


is exact too. Why? My first guess was that we can tensoring the $R\_i$ by $- \otimes\_B A$ via the right inverse $\psi$ from above and exploit then a splitting property of $A \to B$ for above remark,
in order to show somehow that maybe $R\_1 \to R\_2$ has left inverse (or $R\_2 \to R\_3$ right inverse), but I not know how or if it really holds.



Exactness of functor $ Et(B) \to \operatorname{(Ab)}, \ C \mapsto \mathcal{F}(C) $ (Etale Cohomology and the Weil Conjecture by Freitag, Kiehl )
I think they meant to say that the functor $\mathcal F \mapsto \mathcal F(A)$ from sheaves on $A$ to abelian groups is exact.


The proof is:


If $\mathcal F\_2\to \mathcal F\_3 \to 0$ is an exact sequence, we need to show $\mathcal F\_2(A) \to \mathcal F\_3(A)$ is surjective.


Let $s \in \mathcal F\_3(A)$ be a section. By definition of a surjective map of sheaves, there exists a covering $B$ of $A$ such that the pullback of $s$ to $B$ lies in the image of $\mathcal F\_2(B) \to \mathcal F\_3(B)$.


By the previous line, we can express $A$ as an open subset of $B$, so the pullback of $s$ to $\mathcal F\_3(A)$, which is $s$ again, lies in the image of $\mathcal F\_2(A) \to \mathcal F\_3(A)$, as desired.


