### 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.

Original language | English |
---|---|

Pages (from-to) | 171-200 |

Number of pages | 30 |

Journal | Algebras and Representation Theory |

Volume | 4 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 2001 |

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### Keywords

- Cohen-Macaulay approximation
- G-dimension
- Tate-Vogel homology

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Tate-Vogel completions of half-exact functors.** / Yoshino, Yuji.

Research output: Contribution to journal › Article

*Algebras and Representation Theory*, vol. 4, no. 2, pp. 171-200. https://doi.org/10.1023/A:1011437901466

}

TY - JOUR

T1 - Tate-Vogel completions of half-exact functors

AU - Yoshino, Yuji

PY - 2001/6

Y1 - 2001/6

N2 - 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 (Sn and sn), we define F∨(X) = lim SnSn F(X) and F∧(X) = lim Sn 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 = ExtRi (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.

AB - 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 (Sn and sn), we define F∨(X) = lim SnSn F(X) and F∧(X) = lim Sn 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 = ExtRi (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.

KW - Cohen-Macaulay approximation

KW - G-dimension

KW - Tate-Vogel homology

UR - http://www.scopus.com/inward/record.url?scp=0035361133&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035361133&partnerID=8YFLogxK

U2 - 10.1023/A:1011437901466

DO - 10.1023/A:1011437901466

M3 - Article

AN - SCOPUS:0035361133

VL - 4

SP - 171

EP - 200

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

SN - 1386-923X

IS - 2

ER -