Tate homology with respect to cotorsion pairs

ABSTRACT

Given two complete hereditary cotorsion pairs (𝒬,ℛ) and $({\mathcal{𝒬}}^{\prime },{\mathcal{ℛ}}^{\prime })$ in the category of modules which satisfy the conditions 𝒬^′⊆𝒬, $\mathcal{𝒬}\cap \mathcal{ℛ}={\mathcal{𝒬}}^{\prime }\cap {\mathcal{ℛ}}^{\prime }$, and ℛ^′ is enveloping, a kind of Tate homology ${\widehat{Tor}}_{\ast }^{\mathcal{𝒬}\mathcal{ℳ}}$ of modules is introduced and investigated in this article based on complete $({\mathcal{𝒬}}^{\prime }\cap {\mathcal{ℛ}}^{\prime })$-resolutions. It is shown that a module admits complete $({\mathcal{𝒬}}^{\prime }\cap {\mathcal{ℛ}}^{\prime })$-resolutions precisely when it has finite 𝒬-projective dimension. In particular, an Avramov–Martsinkovsky type exact sequence, which is natural in both variables, is constructed to connect such Tate homology functors and relative homology functors. Applications given for the subcategory 𝒢ℱ of Gorenstein flat modules go in three directions. The first is to improve the Avramov–Martsinkovsky type exact sequence appeared in Liang’s work [Tate homology of modules of finite Gorenstein flat dimension. Algebr. Represent. Theory 16:1541–1560 (2013)] by showing that it is indeed natural in both variables. The second is to characterize the finiteness of Gorenstein flat (resp., flat) dimension by using vanishing of Tate homology functors ${\widehat{Tor}}_{\ast }^{\mathcal{𝒢}\mathcal{ℱ}\mathcal{ℳ}}(-,-)$. Finally, by virtue of a conclusion provided by Enochs et al., some balance results are established for ${\widehat{Tor}}_{\ast }^{\mathcal{𝒢}\mathcal{ℱ}\mathcal{ℳ}}$.

KEYWORDS:

• Complete resolution
• cotorsion pair
• finite dimension
• Gorenstein flat module
• Tate homology

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D05
• 13D07
• 18G15
• 18G25

Acknowledgements

The authors would like to thank the referee for his/her helpful comments and suggestions. The authors are indebted to Professor Driss Bennis for helpful discussions on Gorenstein homological dimensions.