Redshift conjecture
In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory [math]\displaystyle{ K(R) }[/math] 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 [math]\displaystyle{ E_{\infty} }[/math]-ring spectra, after Hahn and Wilson did so earlier in the case of the truncated Brown-Peterson spectra [math]\displaystyle{ BP\langle{n}\rangle }[/math].[3]
References
- ↑ Lawson, Tyler (2013). "Future directions". Talbot 2013: Chromatic Homotopy Theory. MIT Talbot Workshop. https://math.mit.edu/events/talbot/2013/19-Lawson-thefuture.pdf.
- ↑ Rognes, John (2000). "Algebraic K-theory of finitely presented ring spectra". Oberwolfach talk. https://www.mn.uio.no/math/personer/vit/rognes/papers/red-shift.pdf.
- ↑ Burklund, Schlank, Yuan (2022). The Chromatic Nullstellensatz
- Notes
- Ausoni, C.; Rognes, J. (2008). "The chromatic red-shift in algebraic K-theory". Enseign. Math. 54 (2): 9–11. https://ncatlab.org/nlab/files/AusoniRognesRedShiftInAlgebraicKTheory.pdf.
- Westerland, C. (2017). "A higher chromatic analogue of the image of J". Geometry & Topology 21 (2): 1033–93. doi:10.2140/gt.2017.21.1033.
- 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. Algebra and Applications. 18. Springer-Verlag. p. 313 (or 301). ISBN 978-1447143932. http://folk.uib.no/nmabd/b/b.pdf.
External links
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