Weights for etale cohomology: why does Deligne's definition work? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:52:07Z http://mathoverflow.net/feeds/question/110179 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110179/weights-for-etale-cohomology-why-does-delignes-definition-work Weights for etale cohomology: why does Deligne's definition work? Mikhail Bondarko 2012-10-20T19:03:19Z 2012-10-21T10:27:50Z <p>For a field $K$ and a variety $X/K$ (whose characteristic could be $0$) I need a 'simple' explanation for the (Deligne's) method of defining weights of the $l$-adic etale cohomology of $\overline{X}$ (the base change of $X$ to the algebraic closure of $K$). Which 'complicated' statements does one need to define and study weights, and what statements here could be proved 'easily' (using basic properties of etale cohomology)? What is the best reference for obtaining an 'understanding' of these things (I prefer reading in English and in Russian:))? </p> <p>Upd. I know some references on the subject (Weil II, Kiehl-Weissauer? SGA IV3, SGAVII2); yet it is difficult to understand which parts of these books contain the information I need. Does there exist any 'guide' to any of these texts? </p> <p>On the other hand, "Weights in arithmetic geometry" by Jannsen is too short.</p> http://mathoverflow.net/questions/110179/weights-for-etale-cohomology-why-does-delignes-definition-work/110181#110181 Answer by Timo Keller for Weights for etale cohomology: why does Deligne's definition work? Timo Keller 2012-10-20T19:13:34Z 2012-10-20T19:22:39Z <p>Complicated (the special case <code>$f: X \to \mathbf{F}_q$</code> proper smooth is Weil I!): Let $\mathcal{F}$ be mixed of weight $\leq i$. Then $R^q\pi_!\mathcal{F}$ is mixed of weight $\leq q+i$ (see Deligne, Weil II, Théorème 1 (3.3.1) or Kiehl-Weissauer, Theorem I.7.1, strengthened in I.9.3)</p> http://mathoverflow.net/questions/110179/weights-for-etale-cohomology-why-does-delignes-definition-work/110187#110187 Answer by Niels for Weights for etale cohomology: why does Deligne's definition work? Niels 2012-10-20T20:25:51Z 2012-10-20T20:25:51Z <p>It seems that your question is not well defined unless $K$ is finitely generated over its prime field. </p> <p>See for instance</p> <p>Jannsen, Uwe Weights in arithmetic geometry. Jpn. J. Math. 5 (2010), no. 1, 73–102. <a href="http://arxiv.org/abs/1003.0927" rel="nofollow">http://arxiv.org/abs/1003.0927</a> or <a href="http://www.springerlink.com/content/207j13t274004070/" rel="nofollow">http://www.springerlink.com/content/207j13t274004070/</a></p> <p>and also (this is in French)</p> <p>Deligne, Pierre Poids dans la cohomologie des variétés algébriques. Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pp. 79–85 Canad. Math. Congress, Montreal, Que., 1975. <a href="http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0079.0086.ocr.pdf" rel="nofollow">http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0079.0086.ocr.pdf</a></p>