Gorenstein cohomology in abelian categories

TL;DR

This paper explores Gorenstein cohomology in abelian categories, establishing a balance theorem for relative cohomology and introducing a notion of perfection, thereby unifying previous results by Holm and the authors.

Contribution

It develops a new framework for relative Gorenstein cohomology in abelian categories using Auslander-Buchweitz approximations and proves a unifying balance theorem.

Findings

Comparison maps between cohomology functors are isomorphisms

A new notion of perfection is introduced in this context

The main theorem generalizes previous results by Holm and others

Abstract

We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context. Our main theorem is a balance result for relative cohomology that simultaneously recovers theorems of Holm and the current authors as special cases.