On the class of Benson's cofibrant modules

Abstract

In this paper, we examine the class of cofibrant modules over a group algebra $kG$, that were defined by Benson in [2]. We show that this class is always the left-hand side of a complete hereditary cotorsion pair in the category of $kG$-modules. It follows that the class of Gorenstein projective $kG$-modules is special precovering in the category of $kG$-modules, if $G$ is contained in the class ${\scriptstyle{{\bf LH}}}\mathfrak{F}$ of hierarchically decomposable groups defined by Kropholler in [20] and $k$ has finite weak global dimension. It also follows that the obstruction to the equality between the classes of cofibrant and Gorenstein projective $kG$-modules can be described, over any group algebra $kG$, in terms of a suitable subcategory of the stable category of Gorenstein projective $kG$-modules.