nLab smash product theorem

Statement

A

For $X$ a homotopy type/spectrum and for all $n$, there is a homotopy pullback

$$\array{ L_{E(n)}X &\longrightarrow& L_{K(n)}X \\ \downarrow && \downarrow \\ L_{E(n-1)}X &\longrightarrow& L_{E(n-1)}L_{K(n-1)}X } \,,$$

where $L_{K(n)}$ denotes the Bousfield localization of spectra at $n$th Morava K-theory and similarly $L_{E(n)}$ denotes localization at Morava E-theory.

(Lurie 10, lect 23, theorem 4)

This implies that for understanding the chromatic tower of any spectrum $X$, it is in principle sufficient to understand all its “chromatic pieces” $L_{K(n)} X$. This is the subject of chromatic homotopy theory.

B

Let $E$ be a ring spectrum and $X$ an arbitrary spectrum. Suppose that there exists an integer $s \geq 1$ such that, for every finite spectrum $F$, the $E$-based Adams spectral sequence for $X \otimes F$ has $E^{p,q}_s$ for $p \geq s$.

If $E$ and $X$ are moreover a p-local spectra then $L_E X$ is a smashing localization.

(Lurie 10, lect 31)

References

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010,

  Lecture 23 The Bousfield Classes of $E(n)$ and $K(n)$ (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 30 Localizations and the Adams-Novikov spectral sequence (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 31 The smash product theorem (pdf)