Resolutions and homological dimensions of DG-modules

Abstract: Recently, Yekutieli introduced projective dimension and injective dimension of DG-modules by generalizing the characterization of projective dimension and injective dimension of ordinary modules by vanishing of Ext-group. In this paper, we introduce DG-version of projective resolution and injective resolution for DG-modules over a connective DG-algebra which are different from known DG-version of projective and injective resolutions. An important feature of these resolutions is that, roughly speaking, the "length" of these resolutions give projective or injective dimensions. We show that these resolutions allows us to investigate basic properties of projective and injective dimensions of DG-modules. As an application we introduce the global dimension of a connective DG-algebra and show that finiteness of global dimension is derived invariant.