group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
In chromatic homotopy theory the redshift conjecture is a conjecture about the nature of the algebraic K-theory spectrum $K(R)$ of an E-infinity ring $R$. Roughly, its say that $K(R)$ has chromatic level one higher than $R$ has.
The conjecture was originally formulated by John Rognes (Rognes 99, Rognes 00).
chromatic level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|
0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |
0th Morava K-theory | $K(0)$ | ||
1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |
first Morava K-theory | $K(1)$ | ||
first Morava E-theory | $E(1)$ | ||
2 | elliptic cohomology | elliptic spectrum $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
3 …10 | K3 cohomology | K3 spectrum | |
$n$ | $n$th Morava K-theory | $K(n)$ | |
$n$th Morava E-theory | $E(n)$ | BPR-theory | |
$n+1$ | algebraic K-theory applied to chrom. level $n$ | $K(E_n)$ (red-shift conjecture) | |
$\infty$ | complex cobordism cohomology | MU | MR-theory |
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 $K(KU)$ of the complex K-theory spectrum $KU$ (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