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Chromatic redshift conjecture for algebraic K-theory

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Let AA be an E∞\mathbf{E}_\infty-ring, and fix a prime pp. Using the telescopes T(i)T(i) at pp, say that AA has height exactly nn when it has height at most nn and height at least nn, where height at most nn means T(i)⊗A≃0T(i)\otimes A\simeq 0 for all i>ni>n, and height at least nn means T(n)⊗A≄0T(n)\otimes A\not\simeq 0.

Chromatic redshift conjecture. If AA has height exactly nn, then K⁡(A)\operatorname{K}(A) is an E∞\mathbf{E}_\infty-ring of height exactly n+1n+1.

This is a weak formulation of the proposed general chromatic redshift phenomenon, motivated by the computation of Ausoni and Rognes showing that algebraic K-theory raises the height of connective complex K-theory from 11 to 22. The statement is open in the generality asserted here.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Chromatic redshift conjecture for algebraic K-theory

    Let RR be an \EE1\EE_1-ring spectrum. Let \Spch⊆\Sp\Spch\subseteq\Sp be the full subcategory generated under colimits and desuspensions from the telescopes T(n)T(n) of vnv_n-self maps of finite spectra of type nn, for n=0,1,…,∞n=0,1,\dots,\infty, with T(∞):=\FFpT(\infty):=\FF_p. Chromatic redshift conjecture. If R∈\SpchR\in\Spch, then

    K(R)∈\Spch.K(R)\in\Spch.

    This conjecture predicts that algebraic K-theory preserves the class of spectra controlled by chromatic telescopes, and is motivated by known examples involving K-theory of algebras over the integers and truncated Brown–Peterson spectra. Its general validity is presented as an optimistic expectation and remains open.

    source: Shachar Carmeli, “On the Non-Existence of J-Homomorphisms of Higher Height”, arXiv:2309.12734 (2023).

References

Primary source

Jordan Levin, “Chromatic Purity in Hermitian K-Theory at p=2”, arXiv:2501.09633 (2025).

algebraic-k-theorychromatic-homotopy-theoryring-spectrastable-homotopy-theory
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