Let $E$ be a spectrum. Then $E$ determines an idempotent localization functor $L_E: \mathrm{Sp} \to \mathrm{Sp}$ sending each spectrum to its $E$-localization. The functor $L_E$ generally does not commute with homotopy colimits. (It does send homotopy colimits in spectra to homotopy colimits in $E$-local spectra, though.) When $L_E$ commutes with homotopy colimits, then it is (by a formal argument) equivalent to smashing with the $E$-local sphere $L_E S$ and the localization functor is said to be smashing.

When $E$ is a Moore space $SG$ for $G$ torsion-free, $L_E$ can be described via "arithmetic" localization: the homotopy groups get localized in the sense of commutative algebra. These are smashing localizations. But there are other examples. A (deep) theorem of Hopkins and Ravenel states that localization with respect to Morava $E$-theory $E_n$ is a smashing localization as well. (By contrast, localization with respect to torsion things tends to involve arithmetic completion, which is not smashing.)

Is it known whether there are additional examples of smashing localizations (in the $p$-local category)?


Finite localizations, as defined by Miller ("Finite localizations", Boletin de la Sociedad Matematica Mexicana 37 (1992), 383–390; preprint here) are also smashing localizations. The finite localization away from the thick subcategory of objects of type $n+1$ is sometimes written as $L_n^f$ or $L\prime_n$ (unless I've messed up the indexing). One formulation of the telescope conjecture is that every smashing localization is a finite localization; in particular, if you write $L_n$ for localization with respect to $E_n$, then a consequence would be that $L_n = L_n^f$.


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