Homotopy category of projective complexes and complexes of Gorenstein projective modules

TL;DR

This paper investigates the homotopy categories of projective, Gorenstein projective, and injective complexes over rings, establishing generation properties, adjoint functors, and equivalences, especially for rings with dualising complexes.

Contribution

It proves that the homotopy category of projective complexes is well generated and that Gorenstein projective complexes are precovering over certain rings, also establishing categorical equivalences.

Findings

Homotopy category of projective complexes is well generated.

Gorenstein projective complexes are precovering over certain rings.

Existence of adjoint functors and categorical equivalences for rings with dualising complexes.

Abstract

Let $R$ be a ring with identity and $\C(R)$ denote the category of complexes of $R$-modules. In this paper we study the homotopy categories arising from projective (resp. injective) complexes as well as Gorenstein projective (resp. Gorenstein injective) modules. We show that the homotopy category of projective complexes over $R$, denoted $\KPC$, is always well generated and is compactly generated provided $\KPR$ is so. Based on this result, it will be proved that the class of Gorenstein projective complexes is precovering, whenever $R$ is a commutative noetherian ring of finite Krull dimension. Furthermore, it turns out that over such rings the inclusion functor $\iota : \K(\RGPrj)\hookrightarrow \KR$ has a right adjoint $\iota_{\rho}$, where $\K(\RGPrj)$ is the homotopy category of Gorenstein projective $R$ modules. Similar, or rather dual, results for the injective (resp. Gorenstein injective) complexes will be provided. If $R$ has a dualising complex, a triangle-equivalence between homotopy categories of projective and of injective complexes will be provided. As an application, we obtain an equivalence between the triangulated categories $\K(\RGPrj)$ and $\K(\RGInj)$, that restricts to an equivalence between $\KPR$ and $\KIR$, whenever $R$ is commutative, noetherian and admits a dualising complex.