nLab telescopic localization

Definition

For some prime number $p$, let $X$ be a finite p-local spectrum. By the periodicity theorem this carries a $v_k$-self map

$$f \;\colon\; \Sigma^k X \longrightarrow X \,.$$

The telescopic localization of $X$ is the homotopy colimit $X[f^{-1}]$ of the direct sequence

$$X \stackrel{\Omega^k f}{\longrightarrow} \Sigma^{-k} X \stackrel{\Omega^{2k} f}{\longrightarrow} \Sigma^{-2k} X \stackrel{}{\longrightarrow} \cdots \,.$$

For $X$ in addition of type $\gt n$, this is a Bousfield localization of spectra. (Lurie 10, prop. 1)

References

Lecture notes include

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010 , Lecture 28 Telescopic localization (pdf)