nLab
red-shift conjecture

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Idea

In chromatic homotopy theory the redshift conjecture is a conjecture about the nature of the algebraic K-theory spectrum K(R)K(R) of an E-infinity ring RR. Roughly, its say that K(R)K(R) has chromatic level one higher than RR has.

The conjecture was originally formulated by John Rognes (Rognes 99, Rognes 00).

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

References

The conjecture originates with

  • John Rognes, Algebraic K-theory of finitely presented ring spectra, lecture at Schloss Ringberg, Germany, January 1999 (pdf)

  • John Rognes, Algebraic K-theory of finitely presented ring spectra, Oberwolfach talk September 2000 (OWF abstract pdf)

The conjecture appears published in

An outlook as of 2013 is in

Previous work motivating the conjecture was the study of the algebraic K-theory K(KU)K(KU) of the complex K-theory spectrum KUKU (also thought of as the classifying space for BDR 2-vector bundles) in

which was motivated by the desire to turn topological K-theory into “a form of” elliptic cohomology by a kind of categorification.

See also

Revised on December 3, 2017 09:43:16 by Urs Schreiber (178.6.238.237)