  The cohomology algebra of the canonical bundle of a compact K\"ahler manifold
is naturally viewed as a module over an exterior algebra. We use the
Bernstein-Gel'fand-Gel'fand correspondence, together with Generic Vanishing
theory, in order to understand the regularity properties of this module. We
also relate it to the infinitesimal theory of the canonical linear series
inside paracanonical space. Finally, we apply vector bundle methods on the
polynomial ring side to obtain inequalities for the holomorphic Euler
characteristic and the Hodge numbers of compact K\"ahler manifolds without
irregular fibrations.
