nLab smashing localization

Idea

If a Bousfield localization of spectra $L_E$ at a spectrum $E$ preserves all direct sums, then it is given by smash product with the $E$-localization of the sphere spectrum

$$L_E X \simeq X \wedge L_E S$$

and is hence called a smashing localization.

Smashing localizations hence in particular

1. preserve (∞,1)-colimits;

2. are monoidal (∞,1)-functors except possibly for preservation of the tensor unit.

see e.g. (GGN 13, p. 8) for discussion.

Examples

• rationalization is smashing: $L_{\mathbb{Q}} \simeq (-) \wedge H \mathbb{Q}$ (e.g Bauer 11, example 1.7 (4))

• For $G$ a group without torsion, then localization at the Moore spectrum $E = S G$ (in particular p-localization) is smashing (see at Bousfield localization of spectra here).

• Localization with respect to Morava E-theory is smashing (Hopkins-Ravenel).

• “finite localizations” are smashing (Miller 92)

Counter-examples

• p-completion is not smashing

Related concepts

• smash product theorem

References

• Jacob Lurie, Chromatic Homotopy Theory Lecture notes, Lecture 22 Morava E-theory and Morava K-theory (pdf)

• Miller, Finite localizations, Boletin de la Sociedad Matematica Mexicana 37 (1992), 383–390 (HopfArchive)

• David Gepner, Moritz Groth, Thomas Nikolaus, Universality of multiplicative infinite loop space machines (arXiv:1305.4550)

• Tilman Bauer, Bousfield localization and the Hasse square (2011) (pdf, pdf), chapter 6 in: Christopher Douglas, John Francis, André Henriques, Michael Hill (eds.), Topological Modular Forms, Mathematical Surveys and Monographs Volume 201, AMS 2014 (ISBN:978-1-4704-1884-7)

• Denis Nardin, section 3.2 of Stability and distributivity over orbital ∞-categories, 2012 (pdf)