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The completion tower of a nonunital commutative ring is a classicalconstruction in commutative algebra. In the setting of structured ring spectraas modeled by algebras over a spectral operad, the analogous construction isthe homotopy completion tower. The purpose of this brief note is to show thatlocalization with respect to the Johnson-Wilson spectrum $E(n)$ commutes withthe terms of this tower.

On the chromatic localization of the homotopy completion tower for $\mathcal{O}$-algebras