Criteria for vanishing of Tor over complete intersections

We exploit properties of Dao's eta-pairing (see Trans. Amer. Math. Soc. 365:6 (2013), 2803-2821), as well as techniques of Huneke, Jorgensen, and Wiegand (J. Algebra 238:2 (2001), 684-702), to study the vanishing of Tor(i)(M,N) for finitely generated modules M, N over complete intersections. We prove vanishing of Tor(i)(M,N) for all i >= 1 under depth conditions on M, N, and M circle times N. Our arguments improve a result of Dao and establish a new connection between the vanishing of Tor and the depth of tensor products.