  Let $(R,m)$ be a complete local ring of positive dimension, which contains a
separably closed coefficient field of prime characteristic. Using a vanishing
theorem of Peskine-Szpiro, Lyubeznik proved that every element of the local
cohomology module $H^1_m(R)$ is killed by an iteration of the Frobenius map if
and only if $R$ has dimension at least two and its punctured spectrum is
connected in the Zariski topology. We give a simple proof of this theorem and
of a variation which, more generally, yields the number of connected
components.
