  Given an algebraic stack $X$, one may compare the derived category of
quasi-coherent sheaves on $X$ with the category of dg-modules over the dg-ring
of functions on $X$. We study the analogous question in stable homotopy theory,
for derived stacks that arise via realizations of diagrams of Landweber-exact
homology theories. We identify a condition (quasi-affineness of the map to the
moduli stack of formal groups) under which the two categories are equivalent,
and study applications to topological modular forms. In particular, we provide
new examples of Galois extensions of ring spectra and vanishing results about
Tate spectra.
