  We give sharp bounds on the vanishing of the cohomology of a tensor product
of vector bundles on the n-dimensional projective space in terms of the
vanishing of the cohomology of the factors. For this purpose we introduce
regularity indices generalizing the Castelnuovo-Mumford regularity. As an
application we give a sufficient condition for a vector bundle to have an
unobstructed deformation theory that depends only on the cohomology table of
the bundle. We construct complete families of bundles with such cohomology
tables.
