Abelian model structures and Ding homological dimensions

Year 2016, Volume: 45 Issue: 5, 1461 - 1474, 01.10.2016

Chongqing Wei Limin Wang Zhongkui Liu

Abstract

Let $R$ be an $n$-FC ring. For $0<t\leq n$, we construct a new abelian model structure on $R$-Mod, called the Ding $t$-projective ($t$-injective) model structure. Based on this, we establish a bijective correspondence between $dg$-$t$-projective ($dg$-$t$-injective) $R$-complexes and Ding $t$-projective ($t$-injective) $A$-modules under some additional conditions, where $A=R[x]/(x^2)$. This gives a generalized version of the bijective correspondence established in [14] between $dg$-projective ($dg$-injective) $R$-complexes and Gorenstein projective (injective) $A$-modules. Finally, we show that the embedding functors $K(\mathcal{D} \mathcal{P})\rightarrow K$ ($R$-Mod) and $K(\mathcal{D} \mathcal{J})\rightarrow K$ ($R$-Mod) have right and left adjoints respectively, where $K(\mathcal{D} \mathcal{P})$ ($K(\mathcal{D} \mathcal{J})$) is the homotopy category of complexes of Ding projective (injective) modules, and $K$ ($R$-Mod) denotes the homotopy category. 

Keywords

model structures, Ding $t$-projective (injective) modules, $dg$-$t$-projective (injective) complexes, adjoint functors

References

• [14] Gillespie J., Hovey M.: Gorenstein model structures and generalized derived categories. Submitted to Proceedings of the Edinburgh Mathematical Society.