Tate–Vogel Completions of Half-Exact Functors

Abstract

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 S n ), we define F ∨(X)=lim → S n S n F(X) and F ∧(X)=lim ← S n S n 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 R i (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.