In this paper, we are concerned with Tate homology tor ̂ of modules of finite Gorenstein FI-flat dimension. We determine connections between Tate, relative and absolute homology. Furthermore, we discuss Tate homology tor ̂ of a FI-cotorsion modules of finite Gorenstein FI-flat dimension over right coherent rings. © 2016, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore.