Let K be an algebraically closed field of characteristic p > 0. We apply a
theorem of C. Han to give an explicit description for the weak Lefschetz
property of the monomial Artinian complete intersection A =
K[X,Y,Z]/(X^d,Y^d,Z^d) in terms of d and p. This answers a question of J.
Migliore, R. M. Miro-Roig and U. Nagel and, equivalently, characterizes for
which characteristics the rank-2 syzygy bundle Syz(X^d,Y^d,Z^d) on PP^2
satisfies the Grauert-Muelich theorem.