symmetric monoidal (∞,1)-category of spectra
For each prime and for each natural number there is a Bousfield localization of spectra
where is the th Morava E-theory (for the given prime ), the th chromatic localization. These arrange into the chromatic tower which for each spectrum is of the form
The homotopy fibers of each stage of the tower
is called the th monochromatic layer of .
The chromatic convergence theorem states mild conditions under which the homotopy limit over this tower is the -localization
of .
Since moreover is the homotopy fiber product
(see at smash product theorem) it follows that in principle one can study a spectrum by understanding all its “chromatic pieces” . This is the topic of chromatic homotopy theory.
The spectral sequence of a filtered stable homotopy type associated with the chromatic tower (regarded as a filtered object in an (infinity,1)-category) is the chromatic spectral sequence (Wilson 13, section 2.1.2)
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010,
Lecture 29 Telescopic vs -localization (pdf)
Dylan WilsonSpectral Sequences from Sequences of Spectra: Towards the
Spectrum of the Category of Spectra_ lecture at 2013 Pre-Talbot Seminar (pdf)
Last revised on November 18, 2013 at 21:18:42. See the history of this page for a list of all contributions to it.