Finiteness of étale Cohomology Groups

Question

Mr. Milne, in "Étale Cohomology", gives the following proposition (p.224, Corollary VI.2.8):

Proposition: Let $F$ a constructible sheaf on $X_{et}$, the small étale site of $X$, $X$ proper over a field $k$. Then $H^{i}(X,F)$ is finite for $i\geq0$. (false?)

He deduces it via Hochschild-Serre from the statement, that on the big étale site of $X$, constructible sheaves are stable under higher direct images of proper Morphisms (p.223, Theorem VI.2.1).

My Question is: Is there a "basic" proof of the proposition, which doesn't involve other Grothendieck topologies than the small étale sites (and possible the Zariski-topology)?

Thanks!

Edit: Actually, Milne himself states in his course notes (http://www.jmilne.org/math/CourseNotes/lec.html, Remark 17.9) that the proposition is wrong for not seperably closed fields $k$, giving the example of $X=Spec(Q)$ and $F=(Z/2Z)_X$. Moreover, he gives the desired proof in the small étale site for $X$ proper over seperably closed fields (the same notes, 17.5-17.8: Please accept my apologies, if I stole your time..

As a new question arises: Where went the proof in "Étale Cohomology" wrong - or did I misunderstand something?

Answer 1

(This was going to be a comment, but it's too long.)

I don't see how the big étale site appears, even in the proof of cor IV.2.8 of Milne. Seems like he's just base changing to the integral closure of the field but using small étale sites all time. (Though I'm not that familiar with Milne, as I use SGA 4 and 4 1/2 as references.)

Anyway, I think the problem with the proof of corollary IV.2.8 is when he says it just follows from the Hochschild-Serre spectral sequence. I would say the Hochschild-Serre spectral sequence gives you a spectral sequence

$E_2^{pq}=H^p(Gal(k_s/k),H^q(X\otimes k_s,F))\Longrightarrow H^{p+q}(X,F)$

($k_s$ is the separable closure of $k$ as in Milne)

Then Milne IV.2.1 tells you that the groups $H^q(X\otimes k_s,F)$ are finite, but it doesn't follow that their $Gal(k_s/k)$-cohomology is finite too. In fact Milne himself gives a counterexample in the part of his notes that you quote.

(Does that make sense ?)

Answer 2

This mistake appears in James Milne's list of errata at http://www.jmilne.org/math/Books/add/ECPUP.pdf.

Answer 3

$H^i(X,\mathscr{F})$ is finite for $X/k$ proper if $k$ is separably closed or finite (the latter by the Hochschild-Serre spectral sequence and $\mathrm{cd}(\mathbf{F}_q) = 1$).