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edited Sep 21, 2011 at 18:11

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 thisit 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)?

I would like to use this for my Graduation thesisThanks!

Edit: Actually, so if you think thisMilne himself states in his course notes (http://www.jmilne.org/math/CourseNotes/lec.html, Remark 17.9) that the proposition is easy and should be a standard exercisewrong for not seperably closed fields $k$, giving the example of $X=Spec(Q)$ and I should solve it myself$F=(Z/2Z)_X$. Moreover, he gives the desired proof in the small étale site for $X$ proper over seperably closed fields (the same notes, 17.5- tell me17.8: Please accept my apologies, if I will try againstole your time..

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

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

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 $i\geq0$.

He deduces this 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)?

I would like to use this for my Graduation thesis, so if you think this is easy and should be a standard exercise and I should solve it myself - tell me, I will try again.

Thanks!

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?

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asked Sep 21, 2011 at 17:42

Jascha

Finiteness of étale Cohomology Groups

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

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 $i\geq0$.

He deduces this 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)?

I would like to use this for my Graduation thesis, so if you think this is easy and should be a standard exercise and I should solve it myself - tell me, I will try again.

Thanks!