  We study two properties of modules over a local hypersurface $R$: decency and
rigidity. We show that the vanishing of Hochster's function $\theta^R(M,N)$,
known to imply decent intersection, also implies rigidity. We investigate the
vanishing of $\theta^R(M,N)$ to obtain new results about decency and rigidity
over hypersurfaces. We employ a mixture of techniques from Commutative Algebra
and Intersection Theory of algebraic cycles.
