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 |
Exposition in:
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
See also
Tyler Lawson, in section 3 of: The future, Talbot lectures 2013 (pdf)
Benjamin Antieau, Some open problems in the K-theory of ring spectra (pdf)
Previous work motivating the conjecture was the study (see also at iterated algebraic K-theory) 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.
For more see the references at iterated algebraic K-theory.
Last revised on September 4, 2020 at 09:00:55. See the history of this page for a list of all contributions to it.