Vanishing of relative homology and depth of tensor products

For finitely generated modules M and N over a Gorenstein local ring R, one has depthM+depthN=depth(M⊗RN)+depthR, i.e., the depth formula holds, if M and N are Tor-independent and Tate homology Torˆi(M,N) vanishes for all i∈Z. We establish the same conclusion under weaker hypotheses: if M and N are G-relative Tor-independent, then the vanishing of Torˆi(M,N) for all i⩽0 is enough for the depth formula to hold. We also analyze the depth of tensor products of modules and obtain a necessary condition for the depth formula in terms of G-relative homology.