smash product theorem


Stable Homotopy theory

Higher algebra




For XX a homotopy type/spectrum and for all nn, there is a homotopy pullback

L E(n)X L K(n)X L E(n1)X L E(n1)L K(n1)X, \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)L_{K(n)} denotes the Bousfield localization of spectra at nnth Morava K-theory and similarly L E(n)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 XX, it is in principle sufficient to understand all its “chromatic pieces” L K(n)XL_{K(n)} X. This is the subject of chromatic homotopy theory.


Let EE be a ring spectrum and XX an arbitrary spectrum. Suppose that there exists an integer s1s \geq 1 such that, for every finite spectrum FF, the EE-based Adams spectral sequence for XFX \otimes F has E s p,qE^{p,q}_s for psp \geq s.

If EE and XX are moreover a p-local spectra then L EXL_E X is a smashing localization.

(Lurie 10, lect 31)