  Morava $E$-theory $E$ is an $E_\infty$-ring with an action of the Morava
stabilizer group $\Gamma$. We study the derived stack $\operatorname{Spf}
E/\Gamma$. Descent-theoretic techniques allow us to deduce a theorem of
Hopkins-Mahowald-Sadofsky on the $K(n)$-local Picard group, as well as a recent
result of Barthel-Beaudry-Stojanoska on the Anderson duals of higher real
$K$-theories.
