Bounds on depth of tensor products of modules

Let R be a local complete intersection ring and let M and N be nonzero finitely generated R  -modules. We employ Auslander's transpose in the study of the vanishing of Tor and obtain useful bounds for the depth of the tensor product M⊗RNM⊗RN. An application of our main argument shows that, if M is locally free on the punctured spectrum of R  , then either depth(M⊗RN)≥depth(M)+depth(N)−depth(R)depth(M⊗RN)≥depth(M)+depth(N)−depth(R), or depth(M⊗RN)≤codim(R)depth(M⊗RN)≤codim(R). Along the way we generalize an important theorem of D.A. Jorgensen and determine the number of consecutive vanishing of ToriR(M,N) required to ensure the vanishing of all higher ToriR(M,N).