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Let $R$ be a ring with identity and $\C(R)$ denote the category of complexesof $R$-modules. In this paper we study the homotopy categories arising fromprojective (resp. injective) complexes as well as Gorenstein projective (resp.Gorenstein injective) modules. We show that the homotopy category of projectivecomplexes over $R$, denoted $\KPC$, is always well generated and is compactlygenerated provided $\KPR$ is so. Based on this result, it will be proved thatthe class of Gorenstein projective complexes is precovering, whenever $R$ is acommutative noetherian ring of finite Krull dimension. Furthermore, it turnsout 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, atriangle-equivalence between homotopy categories of projective and of injectivecomplexes will be provided. As an application, we obtain an equivalence betweenthe triangulated categories $\K(\RGPrj)$ and $\K(\RGInj)$, that restricts to anequivalence between $\KPR$ and $\KIR$, whenever $R$ is commutative, noetherianand admits a dualising complex.

Homotopy category of projective complexes and complexes of Gorenstein projective modules