Resolutions and homological dimensions of DG-modules

TL;DR

This paper introduces new DG-version of projective and injective resolutions for DG-modules over connective DG-algebras, enabling the study of their homological dimensions and invariants like global dimension.

Contribution

It presents novel DG-resolutions that differ from existing ones, linking their length to homological dimensions and establishing derived invariance of global dimension.

Findings

New DG-resolutions for DG-modules are introduced.

Finiteness of global dimension is shown to be a derived invariant.

Resolutions facilitate investigation of projective and injective dimensions.

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.