  In this paper, motivated by a work of Luk and Yau, and Huneke and Wiegand, we
study various aspects of the cohomological rigidity property of tensor product
of modules over commutative Noetherian rings. We determine conditions under
which the vanishing of a single local cohomology module of a tensor product
implies the vanishing of all the lower ones, and obtain new connections between
the local cohomology modules of tensor products and the Tate homology. Our
argument yields bounds for the depth of tensor products of modules, as well as
criteria for freeness of modules over complete intersection rings. Along the
way, we also give a splitting criteria for vector bundles on smooth complete
intersections.
