On the chromatic localization of the homotopy completion tower for \(\mathcal{O}\)-algebras (Q2169887)

Let \({\mathcal{O}}\) be an operad in spectra. Suppose \(X\) is an \({\mathcal{O}}\)-algebra. \textit{J. E. Harper} and \textit{K. Hess} [Geom. Topol. 17, No. 3, 1325--1416 (2013; Zbl 1270.18025)] constructed a tower \(\tau_n(X)\) of \({\mathcal{O}}\)-algebras under \(X\). This tower is analogous to a completion tower with respect to an ideal. Let \(E\) be a spectrum. Let \(L_E\) be localization with respect to \(E\). Recall that \(L_E\) is called a smashing localization if there is a natural equivalence \(L_E(X)\cong L_E(S)\wedge X\), where \(S\) is the sphere spectrum. The authors' main result says that smashing localization commutes with \(\tau_n\) in the tower of Harper and Hess. Most of the work consists of constructing a good model of \(L_E(S)\) as a commutative ring spectrum. En route, the authors give a self-contained proof that \(\tau_n\) is equivalent to Goodwillie's \(n^{\mathrm{th}}\) Taylor approximation of the identity functor on the category of \({\mathcal{O}}\)-algebras.