Algebraic K-theory of quotient ring spectra

We study the algebraic K-theory of quotients of even ring spectra. Building on Efimov’s work, we prove that the limit of the K-theories of these quotients is the continuous K-theory of nuclear modules. For quotients by elements in the height $n$ chromatic ideal, we show that this agrees with the K-theory of the ring spectrum, after chromatic localization at heights $\ge n+1$.