AI Summary & Key Takeaways
AI InsightFinally, 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.
📌 3 Key Takeaways
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.