nLab monochromatic layer

Idea

For each prime $p \in \mathbb{N}$ and for each natural number $n \in \mathbb{N}$ there is a Bousfield localization of spectra

$$L_n \coloneqq L_{E(n)} \,,$$

where $E(n)$ is the $n$th Morava E-theory (for the given prime $p$). These arrange into the chromatic tower which for each spectrum $X$ is of the form

$$X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X \,.$$

The homotopy fibers of each stage of the tower

$$M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))$$

is called the $n$th monochromatic layer of $X$.

Examples

Chromatic layers of the sphere spectrum

Discussion of the first chromatic layer of the sphere spectrum is due to (Miller-Ravenel-Wilson 77). Review is in (Knudsen 13). This is the image of the J-homomorphism.

Related concepts

• chromatic convergence theorem

• smash product theorem

References

General lecture notes include

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 34 Monochromatic layers (pdf)

Discussion of the chromatic layers of the sphere spectrum is in

• Haynes Miller, Doug Ravenel, Stephen Wilson, Periodic phenomena in the Adams-Novikov spectral sequence (JSTOR)

Lecture notes on this include

• Ben Knudsen, First chromatic layer of the sphere spectrum = homotopy of the $K(1)$-local sphere, talk at 2013 Pre-Talbot Seminar (pdf)

• Johan Konter, The $K(n)$-local sphere, talk at Talbot 2013: Chromatic homotopy theory (pdfS.pdf))

• Drew Heard, Resolutions of the $K(2)$-local sphere, talk at Talbot 2013: Chromatic homotopy theory (pdf)