On the derived category of quasi-coherent sheaves on an Adams geometric stack

Let $\mathsf{X}$ be an Adams geometric stack. We show that $\mathsf{D}({\mathsf{A}}_{\mathsf{qc}}(\mathsf{X}))$, its derived category of quasi-coherent sheaves, satisfies the axioms of a stable homotopy category defined by Hovey, Palmieri and Strickland in 13]. Moreover we show how this structure relates to the derived category of comodules over a Hopf algebroid that determines $\mathsf{X}$.