Picard groups of quotient ring spectra

We develop tools to study Picard groups of quotients of ring spectra by a finitely generated ideal, which we use to show that $\mathrm{Pic}(\mathrm{E}_n/I) = \mathbb{Z}/2$, where $\mathrm{E}_n$ is a Lubin--Tate theory and $I$ is an ideal generated by suitable powers of a regular sequence. We apply this to obtain spectral sequences computing Picard groups of $\mathrm{K}(n)$-local generalized Moore algebras, and make some preliminary computations including the height $1$ case.