Smashing Localizations in Equivariant Stable Homotopy

We study how a smashing Bousfield localization behaves under various equivariant functors. We show that the Real Johnson-Wilson theories $E_{\mathbb{R}}(n)$ do not determine smashing localizations except when $n = 0$, and we establish a version of the chromatic convergence theorem for the $L_{E_{\mathbb{R}}(n)}$ chromatic tower. For $G = C_{p^n}$, we construct equivariant Johnson-Wilson theories $E(\mathcal{J})$ corresponding to thick tensor ideals $\mathcal{J}$ in $G$-spectra so that the $E(\mathcal{J})$ do determine smashing localizations. We show that induced localizations upgrade the available norms for an $N_\infty$-algebra, and we determine which new norms appear.

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