Commuting unbounded homotopy limits with Morava K-theory

This paper provides conditions for Morava $K$-theory to commute with certain homotopy limits. These conditions extend previous work on this question by allowing for homotopy limits of sequences of spectra that are not uniformly bounded below. As an application, we prove the $K(n)$-local triviality (for sufficiently large $n$) of the algebraic $K$-theory of algebras over truncated Brown--Peterson spectra, building on work of Bruner--Rognes and extending a classical theorem of Mitchell on $K(n)$-local triviality of the algebraic K-theory spectrum of the integers for large enough $n$.