Some Quillen Equivalences for Model Categories

Let \(({\mathcal {L}}, {\mathcal {A}})\) be a complete duality pair. When R is a commutative ring, we prove a Quillen equivalence induced by a Sharp–Foxby adjunction on R-Mod associated to \(({\mathcal {L}}, {\mathcal {A}})\) between the Gorenstein \(({\mathcal {L}}, {\mathcal {A}})\)-projective and injective model categories, which results in a triangle equivalence between the stable category of Gorenstein \(({\mathcal {L}}, {\mathcal {A}})\)-projective modules and the stable category of Gorenstein \(({\mathcal {L}}, {\mathcal {A}})\)-injective modules. In addition, let R and \(R^{\prime }\) be two (not necessarily commutative) rings. Under some conditions, we investigate the other Quillen equivalence between two Gorenstein \(({\mathcal {L}}, {\mathcal {A}})\)-projective model categories and prove that two stable categories consisting of all Gorenstein \(({\mathcal {L}}, {\mathcal {A}})\)-projective R-modules and all Gorenstein \(({\mathcal {L}}, {\mathcal {A}})\)-projective \(R^{\prime }\)-modules respectively are triangle equivalent by Frobenius functors.