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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.

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    $\begingroup$ 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 $\endgroup$
    – Dan Petersen
    Commented Oct 21, 2012 at 11:32
  • $\begingroup$ I remember that there is an explanation in the third volume (on cohomology) of the encyclopaedia of algebraic geometry which is translated from Russian. The English version is published by Springer. $\endgroup$
    – Z. M
    Commented Mar 27, 2022 at 21:25
  • $\begingroup$ Oh, Dan has mentioned this above. $\endgroup$
    – Z. M
    Commented Mar 27, 2022 at 21:26

2 Answers 2

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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)

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    $\begingroup$ 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? $\endgroup$
    – Mikhail Bondarko
    Commented Oct 20, 2012 at 19:59
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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. https://arxiv.org/abs/1003.0927 or https://doi.org/10.1007/s11537-010-0947-4

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. https://web.archive.org/web/20120713124550/http://www.mathunion.org/ICM/ICM1974.1/Main/icm1974.1.0079.0086.ocr.pdf

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  • $\begingroup$ About the first reference: do you know where I can find some details on the isomorphism $b_{\eta,s}$ in the formula (2.3)? $\endgroup$
    – Mikhail Bondarko
    Commented Oct 20, 2012 at 21:03
  • $\begingroup$ 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 $\endgroup$
    – Niels
    Commented Oct 21, 2012 at 7:52
  • $\begingroup$ 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? $\endgroup$
    – Mikhail Bondarko
    Commented Oct 21, 2012 at 9:29
  • $\begingroup$ Since the link to the book mentioned in a comment above by Niels at springerlink.com is broken, I'll just mention that it is also available at doi:10.1007/BFb0070714. $\endgroup$
    – The Amplitwist
    Commented Apr 14, 2022 at 14:37

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