  In this paper we exploit properties of Dao's eta-pairing as well as
techniques of Huneke, Jorgensen, and Wiegand to study the vanishing of
Tor_i(M,N) of finitely generated modules M, N over complete intersections. We
prove vanishing of Tor_i(M, N) for all positive integers i under depth
conditions on M, N, and their tensor product. Our arguments improve a result of
Dao and establish a new connection between the vanishing of Tor and the depth
of tensor products.
