Tate-Vogel and relative cohomologies of complexes with respect to cotorsion pairs

TL;DR

This paper investigates Tate-Vogel and relative cohomologies of complexes within the framework of cotorsion pairs, establishing connections with Gorenstein dimensions and providing methods for their computation.

Contribution

It characterizes complexes with finite Gorenstein $ ext{A}$ dimension via complete $ ext{A}$ resolutions and develops techniques for computing related cohomologies.

Findings

Class of complexes with complete $ ext{A}$ resolutions equals those with finite Gorenstein $ ext{A}$ dimension

Provides methods for calculating Tate-Vogel cohomologies of complexes with finite Gorenstein $ ext{A}$ dimension

Explores relationships between Gorenstein $ ext{A}$ dimensions and $ ext{A}$ dimensions for complexes

Abstract

We study Tate-Vogel and relative cohomologies of complexes by applying the model structure induced by a complete hereditary cotorsion pair ($\A$, $\B$) of modules. We show first that the class of complexes admitting a complete $\A$ resolution is exactly the class of complexes with finite Gorenstein $\A$ dimension. This lets us give general techniques for computing Tate-Vogel cohomoloies of complexes with finite Gorenstein $\A$ dimension. As a consequence, relative cohomology groups for complexes with finite Gorenstein $\A$ dimension are investigated. Finally, the relationships between Gorenstein $\A$ dimensions and $\A$ dimensions for complexes are given.