# Reference for comparison of complex and étale cohomology

Hello everybody!

I am looking for a nice DETAIL account of the comparison of étale cohomology and complex cohomology, an alternative reference instead of SGA. Especially on this stuff of "Artin neighbourhoods". Milne's otherwisely great book is a bit sketchy here, SGA is great but overwhelms my little head, I couldn't find it in Freitag-Kiehl at all.

Yes, yes, eventually I would be "happy" to dig through SGA, but we all know that it simplifies things so much having two or three accounts of the same thing to read simultaneously.

Thank you very much.

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Do you want relative version (e.g., higher direct images, not nec. with proper supports), do you want $\ell$-adic versions (literature is bad on this step)? For ordinary cohomology, proof in sga is harder than compact supports (due to excision issues), and relies on resolution. But Berkovich's comparison proof (in torsion case) for his analytic spaces in IHES avoids resolution (forced: wants char. > 0 too!), and actually works in alg. case to give simpler & resolution-free proof. Artin nbhds can also be avoided; I've never paid attention to that stuff. –  BCnrd May 30 '10 at 16:21

So I guess BCnrd's very nice answer refers to

78_5_0.djvu">http://archive.numdam.org/article/PMIHES_1993_78_5_0.djvu

Thm. 7.1.1. on p128 This in turn reduces it to 6.1.1.

where the rigid spaces need to be replaced by classical complex manifolds. Berkovich's text really talks a lot about discs etc., as complex geometers would, and even though I don't know any rigid analytic stuff, the little bit of the proof I've read so far really was quite easily rephraseable for the complex case. Cool Cool! Thank you!

The only thing so far that I find truly mysterious and intranslateable is Berkovich's comments à la "if the valuation of k is trivial, this holds for all schemes..." (see Rmk. 6.3.8) - what would be the complex geometry analogon of a trivial valuation?!?! This simply is some sort of (seemingly useless?) generality complex geometry doesn't consider or what does this mean?

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Kai, it is indeed remarkable how easily Berkovich's argument carries over to the "classical" case. (You may also want to follow up on his inspiration, the expose by Deligne from sga 4.5 which proves important constructibility results for higher direct images and vanishing cycles.) As for Berkovich's insistence on the trivial valuation, despite appearances it actually has substance and isn't vacuous, but has no classical counterpart, so just ignore it (until such day as you learn about Berkovich spaces). –  BCnrd May 30 '10 at 22:39