by Yoshino, Yuji

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.

DOI: 10.1023/A:1011437901466
Print publication date: 6/1/2001
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