Duality pairs and stable module categories

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.