Most calculations of étale cohomology in Milne's book deal with constructible or torsion sheaves. Are there references where the cohomology of varieties with $\mathbf{G}_m$ coefficients are calculated? I'm especially interested in the top dimension $2\mathrm{dim}(X)$ ($+ 1$). I found some calculations in Le groupe de Brauer (In: Dix Exposés sur la Cohomologie des Schémas), but they don't help me.

Edit: Assume $X$ is a variety over a finite field.

  • $\begingroup$ You could as an exercise try to do Kummer theory using etale cohomology. That is essentially the heart of the whole business. $\endgroup$ – Anweshi Jan 26 '10 at 15:56
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    $\begingroup$ According to a story I heard from Serre, after Grothendieck came up with the formalism of sites, etc., very quickly after attending a lecture by Serre, he (=AG) got completely stuck for a year or so because he couldn't compute anything for a curve beyond constant coefficients (using SGA1). No kidding. It was M. Artin who saw how to use Brauer/Tsen/Kummer/etc. to compute more for curves, and then they were off and running. So for a beginner to hear that the idea of relating GL_1-cohomology to other cohomologies on a variety is an "exercise" probably is not a good thing. $\endgroup$ – BCnrd Feb 24 '10 at 2:31

I found calculations in S. Lichtenbaum, Zeta functions of varieties over finite fields at s = 1, Arithmetic and geometry, Vol. I, 173–194 Progr. Math., 35, Birkhauser Boston, Boston, MA, 1983, especially Proposition 2.1 and Theorem 2.2--2.4.

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