Homotopy equivalences and Grothendieck duality over rings with finite Gorenstein weak global dimension

Abstract

Let $R$ be a ring with Gwgldim$(R)<\infty$. We obtain a triangle-equivalence $\mathrm{K}(R\text{-}\mathrm{GProj})\simeq \mathrm{K}(R\text{-}\mathrm{GInj})$ which restricts to a triangle-equivalence $\mathrm{K}(R\text{-}\mathrm{Proj})$ $\simeq \mathrm{K}(R\text{-}\mathrm{Inj})$. This class of rings includes, among others, (left) Gorenstein rings, Ding-Chen rings and the more general Gorenstein $n$-coherent rings ($n\in \mathbb{N}\cup \{\infty \},n\ge 2$). As application, we establish some triangle-equivalences of Grothendieck duality over Ding-Chen rings and Gorenstein $n$-coherent rings.