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 $\mathrm{P}\mathrm{i}\mathrm{c}({\mathrm{E}}_n\mathrm{/}I)=\mathbb{Z}\mathrm{/}2$, 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.