On the class of Benson's cofibrant modules

Abstract

In this paper, we study the class of cofibrant modules over a group algebra $kG$, that were introduced by Benson. We prove that this class is always the left-hand side of a complete hereditary and projective cotorsion pair. We also examine the relation between cofibrant and Gorenstein projective modules and the behaviour of cofibrant modules with respect to the induction functor from subgroups. As an application, we show that the cofibrant cotorsion pair induces a monoidal model structure on the category of $kG$-modules over {\em any} commutative coefficient ring $k$, provided that the group $G$ is obtained from the class of finite groups by iterated applications of Kropholler's operation $\mathbf{LH}$ and Talelli's operation $\Phi$. The corresponding symmetric monoidal homotopy category is equivalent to the stable category of cofibrant $kG$-modules.