  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 $\mathbf{X}$.
