Redshift conjecture

In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory ${\displaystyle K(R)}$ has chromatic level one higher than that of a complex-oriented ring spectrum R.^[1] It was formulated by John Rognes in a lecture at Schloss Ringberg, Germany, in January 1999, and made more precise by him in a lecture at Mathematische Forschungsinstitut Oberwolfach, Germany, in September 2000.^[2] In July 2022, Robert Burklund, Tomer Schlank and Allen Yuan announced a solution of a version of the redshift conjecture for arbitrary ${\displaystyle E_{\infty }}$-ring spectra, after Hahn and Wilson did so earlier in the case of the truncated Brown-Peterson spectra ${\displaystyle BP\langle {n}\rangle }$.^[3]

References

1. Lawson, Tyler (2013). "Future directions" (PDF). Talbot 2013: Chromatic Homotopy Theory. MIT Talbot Workshop.
2. Rognes, John (2000). "Algebraic K-theory of finitely presented ring spectra" (PDF). Oberwolfach talk.
3. Burklund, Schlank, Yuan (2022). The Chromatic Nullstellensatz

Notes

• Ausoni, C.; Rognes, J. (2008). "The chromatic red-shift in algebraic K-theory" (PDF). Enseign. Math. 54 (2): 9–11.
• Westerland, C. (2017). "A higher chromatic analogue of the image of J". Geometry & Topology. 21 (2): 1033–93. arXiv:1210.2472. doi:10.2140/gt.2017.21.1033. S2CID 44643197.

• Burklund, Robert; Schlank, Tomer M.; Yuan, Allen (2022). "The Chromatic Nullstellensatz". arXiv:2207.09929 [math.AT].

Further reading

• Dundas, Bjørn Ian; Goodwillie, Thomas G.; McCarthy, Randy (2012). The Local Structure of Algebraic K-Theory (PDF). Algebra and Applications. Vol. 18. Springer-Verlag. p. 313 (or 301). ISBN 978-1447143932.

External links

• red-shift conjecture at the nLab