  Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb
Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local
cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This
settles a crucial outstanding case of a conjecture of Lyubeznik asserting this
finiteness for local cohomology modules of all regular rings.
