We show that there is an Avramov-Martsinkovsky type exact sequence with Tor, Gtor, and Tor. We prove that if R is a Gorenstein ring, then the modules Tor R n M N , n ≥ 1 can be computed using either a complete resolution of M R or using a complete resolution of R N . We show that over a Gorenstein ring a left R-module N is Gorenstein flat if and only if Gtor R 1 − N = 0. We also show that over commutative Gorenstein rings the modules Tor R n M − can be computed by the combined use of a flat resolution and a Gorenstein flat resolution of M.
We consider a right coherent ring R. We prove that the class of Gorenstein flat complexes is covering in the category of complexes of left R-modules Ch(R).
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