Abstract. In this paper, we show that the homotopy category of N -complexes of projective R-modules is triangle equivalent to the homotopy category of projective T N´1 pRqmodules where T N´1 pRq is the ring of triangular matrices of order N´1 with entries in R. We also define the notions of N -singularity category and N -totally acyclic complexes. We show that the category of N -totally acyclic complexes of finitely generated projective R-modules embeds in the N -singularity category, which is a result analogous to the case of ordinary chain complexes.

The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalise two results of Happel by proving the existence of AR-triangles in Gorenstein derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent and finally study relations between the Gorenstein derived category of a quiver and its image under a reflection functor. Some interesting applications are provided.

In this paper, we study the category of sheaves over an infinite partially ordered set with its natural topological structure. Totally acyclic complexes in this category will be characterized in terms of their stalks. This leads us to describe Gorenstein projective, injective and flat sheaves. As an application, we get an analogue of a formula due to Mitchell, giving an upper bound on the Gorenstein global dimension of such categories. Based on these results, we present situations in which the class of Gorenstein projective sheaves is precovering as well as situations in which the class of Gorenstein injective sheaves is preenveloping.

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