We show that there is an Avramov–Martsinkovsky type exact sequence with , Gtor, and Tor. We prove that if R is a Gorenstein ring, then the modules , n ≥ 1 can be computed using either a complete resolution of MR or using a complete resolution of RN. We show that over a Gorenstein ring a left R-module N is Gorenstein flat if and only if . We also show that over commutative Gorenstein rings the modules can be computed by the combined use of a flat resolution and a Gorenstein flat resolution of M.