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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 $i$. \leq i$. Then$R^q\pi_!\mathcal{F}$is mixed of weight$\leq q+i$(see Deligne, Weil II, Théorème 1 (3.3.1) or Kiehl-Weissauer, Theorem I.7.1, strengthened in I.9.3) 1 Complicated: Let$\mathcal{F}$be mixed of weight$i$. Then$R^q\pi_!\mathcal{F}$is mixed of weight$q+i\$ (see Deligne, Weil II, or Kiehl-Weissauer, Theorem I.7.1, strengthened in I.9.3)