## Tsukuba Journal of Mathematics

### Generalized Tate cohomology

Alina Iacob

#### Abstract

We consider two classes of left $R$-modules, $\mathscr{P}$ and $\mathscr{C}$, such that $\mathscr{P}\subset \mathscr{C}$. If the module $M$ has a $\mathscr{P}$-resolution and a $\mathscr{C}$-resolution then for any module $N$ and $n\geq 0$ we define generalized Tate cohomology modules $\widehat{Ext}_{\mathscr{C},\mathscr{P}}^{n}(M, N)$ and show that we get a long exact sequence connecting these modules and the modules $Ext_{\mathscr{C}}^{n}(M, N)$ and $Ext_{\mathscr{P}}^{n}(M, N)$. When $\mathscr{C}$ is the class of Gorenstein projective modules, $\mathscr{P}$ is the class of projective modules and when $M$ has a complete resolution we show that the modules $\widehat{Ext}_{\mathscr{C},\mathscr{P}}^{n}(M, N)$ for $n\geq 1$ are the usual Tate cohomology modules and prove that our exact sequence gives an exact sequence provided by Avramov and Martsinkovsky. Then we show that there is a dual result. We also prove that over Gorenstein rings Tate cohomology $\widehat{Ext}_{R}^{n}(M, N)$ can be computed using either a complete resolution of $M$ or a complete injective resolution of $N$. And so, using our dual result, we obtain Avramov and Martsinkovsky's exact sequence under hypotheses different from theirs.

#### Article information

Source
Tsukuba J. Math., Volume 29, Number 2 (2005), 389-404.

Dates
First available in Project Euclid: 30 May 2017

https://projecteuclid.org/euclid.tkbjm/1496164963

Digital Object Identifier
doi:10.21099/tkbjm/1496164963

Mathematical Reviews number (MathSciNet)
MR2177019

Zentralblatt MATH identifier
1095.16003

#### Citation

Iacob, Alina. Generalized Tate cohomology. Tsukuba J. Math. 29 (2005), no. 2, 389--404. doi:10.21099/tkbjm/1496164963. https://projecteuclid.org/euclid.tkbjm/1496164963