Tate-Vogel completions of half-exact functors

We provide a general method to construct the Tate-Vogel homology theory for a general half-exact functor with one variable, aiming at a good generalization of Cohen-Macaulay approximations of modules over commutative Gorenstein rings. For a half exact functor F, using the left and right satellites (S_n and sⁿ), we define F^∨(X) = lim S_nSⁿ F(X) and F^∧(X) = lim S_n Sⁿ F(X), and call F^∨ and F^∧ the Tate-Vogel completions of F. We provide several properties of F^∨ and F^∧, and their relations with the G-dimension and the projective dimension of the functor F. A comparison theorem of Tate-Vogel completions with ordinary Tate-Vogel homologies is proved. If F is a half exact functor over the category of R-modules, where R is a commutative Noetherian local ring inspired by Martsinkovsky's works, we can define the invariants ξ(F) and η(F) of F. If F = Ext_Rⁱ (M,), then they coincide with Martsinkovsky's ξ-invariants and Auslander's delta invariants. Our advantage is that we can consider these invariants for any half exact functors. We also compute these invariants for the local cohomology functors.