Duality pairs and stable module categories

TL;DR

This paper develops a general framework for relative homological algebra based on complete duality pairs over commutative rings, extending Gorenstein homological algebra and constructing new abelian model structures.

Contribution

It introduces a unified theory of relative homological algebra from duality pairs, generalizing Gorenstein homological algebra and extending existing model structures.

Findings

Gorenstein homological algebra can be derived from duality pairs over Noetherian rings.

The work extends to general rings using duality pairs of level and absolutely clean modules.

New abelian model structures are constructed from any complete duality pair.

Abstract

Let $R$ be a commutative ring. We show that any complete duality pair gives rise to a theory of relative homological algebra, analogous to Gorenstein homological algebra. Indeed Gorenstein homological algebra over a commutative Noetherian ring of finite Krull dimension can be recovered from the duality pair $(\mathcal{F},\mathcal{I})$ where $\mathcal{F}$ is the class of flat $R$-modules and $\mathcal{I}$ is the class of injective $R$-modules. For a general $R$, the AC-Gorenstein homological algebra of Bravo-Gillespie-Hovey is the one coming from the duality pair $(\mathcal{L},\mathcal{A})$ where $\mathcal{L}$ is the class of level $R$-modules and $\mathcal{A}$ is class of absolutely clean $R$-modules. Indeed we show here that the work of Bravo-Gillespie-Hovey can be extended to obtain similar abelian model structures on $R$-Mod from any a complete duality pair $(\mathcal{L},\mathcal{A})$. It applies in particular to the original duality pairs constructed by Holm-J{\o} rgensen.