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In this paper we are concerned with absolute, relative and Tate Tor modules.In the first part of the paper we generalize a result of Avramov andMartsinkovsky by using the Auslander-Buchweitz approximation theory, and obtaina new exact sequence connecting absolute Tor modules with relative and Tate Tormodules. In the second part of the paper we consider a depth equality, calledthe depth formula, which has been initially introduced by Auslander anddeveloped further by Huneke and Wiegand. As an application of our main result,we generalize a result of Yassemi and give a new sufficient condition implyingthe depth formula to hold for modules of finite Gorenstein and finite injectivedimension.

A study of Tate homology via the approximation theory with applications to the depth formula