nLab telescope conjecture

The telescope conjecture

The telescope conjecture

Idea

In the stable homotopy category of pp-local spectra for some prime number pp, there are two collections of similarly behaved spectra: the Morava K-theories and spectra which are the telescopes of finite spectra under their v nv_n-maps. The former are usually denoted K(n)K(n) and the latter are often denoted T(n)T(n). The telescope conjecture is that localization at K(n)K(n) and localization at T(n)T(n) are (Bousfield) equivalent.

The conjecture is known to be true for n=1n=1. A disproof of this conjecture for n2n\geq 2 using algebraic K-theory has been released by Burklund, Hahn, Levy & Schlank 2023.

References

Proof at height 1:

Disproof at height 2\geq 2:

Last revised on April 20, 2024 at 04:41:50. See the history of this page for a list of all contributions to it.