Descent and finite permutation resolutions for discrete groups

Let G be a discrete group with a finite-dimensional model for the classifying space for proper actions, and let k be a commutative Noetherian ring of finite global dimension. In this setting, we prove that the homotopy category of projective $kG$-modules, the stable module category of $kG$-modules as defined by Mazza-Symonds, and the derived category of permutation $kG$-modules with finite isotropy, admit descent to finite subgroups. As an application, we show that any $kG$-module of type FP_∞ is a retract of a module that admits a finite resolution by finitely generated -permutation modules with finite isotropy, generalizing a result of Balmer-Gallauer.