The Picard group of a commutative ring is an important invariant with wide applications in algebraic geometry and number theory. In chromatic homotopy theory, we study the Picard group groups of the category of $K(n)$-local spectra. Hovey-Strickland showed that those Picard groups admit a profinite topology.

In this talk, we give an infinity-categorical explanation of this topology on $K(n)$-local Picard groups. Building upon Burklund's result on the multiplicative structures of generalized Moore spectra, we prove that the module category over a $K(n)$-local commutative ring spectrum is equivalent to the limit of its base changes by a tower of generalized Moore spectra of type $n$. We also explain how to compute Picard groups of those generalized Moore spectra. This is joint work with Ishan Levy and Guchuan Li.