MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

For ample $L$, Frobenius eventually kills cohomology of $H^i(X, L^{-1})$ for $i < \dim X$ and kills $H^i(X, L)$ for $i > 0$.
(In other words, the natural map $F^e : H^i(X, L) \to H^i(X, L^{p^e})$ has zero target for $e \gg 0$.)