  Let $R$ be a polynomial ring in finitely many variables over the integers,
and fix an ideal $I$ of $R$. We prove that for all but finitely prime integers
$p$, the Bockstein homomorphisms on local cohomology, $H^k_I(R/pR)\to
H^{k+1}_I(R/pR)$, are zero. This provides strong evidence for Lyubeznik's
conjecture which states that the modules $H^k_I(R)$ have a finite number of
associated prime ideals.
