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

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