The Lubin-Tate stack and Gross-Hopkins duality

Morava $E$-theory $E$ is an $E_{\mathrm{\infty }}$-ring with an action of the Morava stabilizer group $\mathrm{\Gamma }$. 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.