The main idea in the proof of Artin's vanishing - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:39:48Z http://mathoverflow.net/feeds/question/109118 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109118/the-main-idea-in-the-proof-of-artins-vanishing The main idea in the proof of Artin's vanishing Mikhail Bondarko 2012-10-08T07:14:40Z 2012-10-08T08:52:27Z <p>Does anybody know an easy explanation of the proof of Artin's vanishing theorem (that the etale cohomology of an affine variety of dimension $n$ over an algebraically closed field vanishes in degrees $>n$, or of any other version of this statement)? I have found some proofs; all of them are step by step, and it is not clear to me which of these steps are the most important ones. So, what is the central idea here?</p> http://mathoverflow.net/questions/109118/the-main-idea-in-the-proof-of-artins-vanishing/109123#109123 Answer by Piotr Achinger for The main idea in the proof of Artin's vanishing Piotr Achinger 2012-10-08T08:52:27Z 2012-10-08T08:52:27Z <p>I was curious myself after learning this result sometime ago from Lazarsfeld's book on positivity (he calls it the Artin-Grothendieck theorem). The corresponding statement for smooth varieties over the complex numbers and singular cohomology (theorem of Andreotti-Frankel) follows from the fact that Morse theory shows that the variety is homotopy equivalent to a CW-complex with no cells in dimensions $>n$. </p> <p>The etale cohomology counterpart works more generally for constructible sheaves. This is probably not very helpful, but here is a sketch of the argument from Lazarsfeld (he uses constructible sheaves in the complex topology, but it should adapt to etale sheaves):</p> <ol> <li>Reduce to the affine space by choosing a finite map $X\to \mathbb{A}^n$ using Noether normalization (already here it is crucial to work with constructible sheaves, not constant sheaves, so we clearly gain something from generalization),</li> <li>Prove the result for $\mathbb{A}^1$,</li> <li>Prove the result for $\mathbb{A}^n$ by induction on $n$ using the Leray spectral sequence. Here the crucial observation is that if we choose a sufficiently generic linear projection $\mathbb{A}^n\to \mathbb{A}^{n-1}$, then the <em>stalks</em> of the higher direct images will compute cohomology on the fibers.</li> </ol> <p>So all in all it is a typical example of <em>devissage</em>, which I usually to dislike but slowly learn to appreciate. I think from the outline it is clear which are the key ideas, but I would still really like to see a conceptual proof.</p>