  We construct p-adic relative cohomology for a family of toric exponential
sums which generalize the classical Kloosterman sums. Under natural hypotheses
such as quasi-homogeneity and nondegeneracy, this cohomology is acyclic except
in the top dimension. Our construction enables sufficiently sharp estimates for
the action of Frobenius on cohomology so that our earlier work may be applied
to the L-functions coming from linear algebra operations on these families to
deduce a number of basic properties.
