Avramov-Martsinkovsky type exact sequences for extriangulated categories

TL;DR

This paper develops a new framework for $\xi$-Gorenstein cohomology within extriangulated categories, establishing balance and exact sequences that connect various cohomology theories.

Contribution

It introduces $\xi$-Gorenstein cohomology via resolutions and coresolutions, and derives Avramov-Martsinkovsky type exact sequences in this generalized setting.

Findings

Established the balance of $\xi$-Gorenstein cohomology.

Derived Avramov-Martsinkovsky type exact sequences.

Explored relationships among different $\xi$-cohomology theories.

Abstract

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we first introduce the $\xi$-Gorenstein cohomology in terms of $\xi$-$\mathcal{G}$projective resolutions and $\xi$-$\mathcal{G}$injective coresolutions, respectively, and then we get the balance of $\xi$-Gorenstein cohomology. Moreover, we study the interplay among $\xi$-cohomology, $\xi$-Gorenstein cohomology and $\xi$-complete cohomology, and obtain the Avramov-Martsinkovsky type exact sequences in this setting.