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

Criteria for vanishing of Tor over Complete Intersections