# 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