Redshift_conjecture Knowpia

In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory $K(R)$ 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 $E_{\infty }$ -ring spectra, after Hahn and Wilson did so earlier in the case of the truncated Brown-Peterson spectra $BP\langle {n}\rangle$ .^[3]

References edit

^ Lawson, Tyler (2013). "Future directions" (PDF). Talbot 2013: Chromatic Homotopy Theory. MIT Talbot Workshop.
^ Rognes, John (2000). "Algebraic K-theory of finitely presented ring spectra" (PDF). Oberwolfach talk.
^ 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.