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.
Best Answer
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)