  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.
