symmetric monoidal (∞,1)-category of spectra
For each prime $p \in \mathbb{N}$ and for each natural number $n \in \mathbb{N}$ there is a Bousfield localization of spectra
where $E(n)$ is the $n$th Morava E-theory (for the given prime $p$), the $n$th chromatic localization. These arrange into the chromatic tower which for each spectrum $X$ is of the form
The homotopy fibers of each stage of the tower
is called the $n$th monochromatic layer of $X$.
The chromatic convergence theorem states mild conditions under which the homotopy limit over this tower is the $p$-localization
of $X$.
Since moreover $L_n X$ is the homotopy fiber product
(see at smash product theorem) it follows that in principle one can study a spectrum $X$ by understanding all its “chromatic pieces” $L_{K(n)} X$. 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)
chromatic level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|
0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |
0th Morava K-theory | $K(0)$ | ||
1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |
first Morava K-theory | $K(1)$ | ||
first Morava E-theory | $E(1)$ | ||
2 | elliptic cohomology | elliptic spectrum $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
3 …10 | K3 cohomology | K3 spectrum | |
$n$ | $n$th Morava K-theory | $K(n)$ | |
$n$th Morava E-theory | $E(n)$ | BPR-theory | |
$n+1$ | algebraic K-theory applied to chrom. level $n$ | $K(E_n)$ (red-shift conjecture) | |
$\infty$ | complex cobordism cohomology | MU | MR-theory |
Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010,
Lecture 29 Telescopic vs $E_n$-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.