# Weights for etale cohomology: why does Deligne's definition work?

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:))?

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?

On the other hand, "Weights in arithmetic geometry" by Jannsen is too short.

• There might be some useful things in the text "Cohomology of algebraic varieties" by Danilov, or rather the parts of it about étale cohomology. It appeared in English translation in an EMS volume, the Russion original can be found here: mi.mathnet.ru/eng/intf124 – Dan Petersen Oct 21 '12 at 11:32

Complicated (the special case $f: X \to \mathbf{F}_q$ 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)

• Thank you! Yet I also need to bound weights from below.:) Certainly, I can apply the Verdier duality to this end; yet is there an 'easier' way? – Mikhail Bondarko Oct 20 '12 at 19:59

It seems that your question is not well defined unless $K$ is finitely generated over its prime field.

See for instance

Jannsen, Uwe Weights in arithmetic geometry. Jpn. J. Math. 5 (2010), no. 1, 73–102. http://arxiv.org/abs/1003.0927 or http://www.springerlink.com/content/207j13t274004070/

and also (this is in French)

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. http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0079.0086.ocr.pdf

• About the first reference: do you know where I can find some details on the isomorphism $b_{\eta,s}$ in the formula (2.3)? – Mikhail Bondarko Oct 20 '12 at 21:03
• You can find all details there Théorie des topos et cohomologie étale des schémas. Tome 3.Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4). Lecture Notes in Mathematics, Vol. 305. springerlink.com/content/n880177446r8 – Niels Oct 21 '12 at 7:52
• I'm sorry; could you give a more precise reference? I know the smooth and proper base change theorems; yet how do they help here? – Mikhail Bondarko Oct 21 '12 at 9:29