nLab red-shift conjecture

Contents

Context

Higher algebra

higher algebra

universal algebra

Contents

Idea

In chromatic homotopy theory the redshift conjecture is a conjecture about the nature of the iterated algebraic K-theory spectrum $K(R)$ of a connective 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) and appeared in Ausoni & Rognes 2008, review in Rognes 2014.

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 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 cohomologyMUMR-theory

References

Exposition in:

The conjecture originates with

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

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

The conjecture appears published in

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 July 22, 2021 at 08:58:34. See the history of this page for a list of all contributions to it.