nLab red-shift conjecture

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.

Related concepts

• BDR 2-vector bundle

• iterated algebraic K-theory

chromatic homotopy theory

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

References

Exposition in:

• John Rognes, Chromatic redshift, lecture notes, MSRI 2014 (arXiv:1403.4838)

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

• Christian Ausoni, John Rognes, The chromatic red-shift in algebraic K-theory, Enseign. Math. (2) 54 (2008), 9-11. (pdf, pdf, doi:10.5169/seals-109873)

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

• Nils Baas, Bjørn Dundas, Birgit Richter, John Rognes, Ring completion of rig categories (arXiv:0706.0531)

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.