  In this paper, we show that the cohomology of a general stable bundle on a
Hirzebruch surface is determined by the Euler characteristic provided that the
first Chern class satisfies necessary intersection conditions. More generally,
we compute the Betti numbers of a general stable bundle. We also show that a
general stable bundle on a Hirzebruch surface has a special resolution
generalizing the Gaeta resolution on the projective plane. As a consequence of
these results, we classify Chern characters such that the general stable bundle
is globally generated.
