The localization sequence for the algebraic \(K\)-theory of topological \(K\) -theory. (English) Zbl 1149.18008

Blumberg, Andrew J.; Mandell, Michael A.

Acta Math. 200, No. 2, 155-179 (2008).

C. Ausoni and J. Rognes [Acta Math. 188, No. 1, 1–39 (2002; Zbl 1019.18008)] described an ambitious program for analyzing the layers in the \(p\)-complete version of Waldhausen’s \(K\)-theory chromatic tower. It was related to the layers of the tower to the \(K\)-theory of Morawa \(E\)-theory ring spectra \(E_n\) and the \(K\)-theory of the \(p\)-completed Johnson-Wilson ring spectra \(E(n)^\wedge_p\). The spectrum \(E(n)\) is not connective, but is formed from the connective spectrum \(BP\langle n\rangle\) by inverting the element \(v_n\) in \(\pi_*(BP\langle n\rangle )\). The Rognes conjecture states that the transfer map \(K(BP\langle n-1\rangle^\wedge_p) \to K(BP\langle n\rangle^\wedge_p)\) and the canonical map \(K(BP\langle n\rangle^\wedge_p) \to K(E(n)^\wedge_p)\) fit into a cofiber sequence in the stable category \[ K(BP\langle n-1\rangle^\wedge_p) \to K(BP\langle n\rangle^\wedge_p) \to K(E(n)^\wedge_p) \to \Sigma K(BP\langle n-1\rangle^\wedge_p). \] In the case \(n=0\), the statement is an old theorem of D. Quillen [Lect. Notes Math. 341, 85–147 (1973; Zbl 0292.18004)], the localization sequence \(K(\mathbb Z/p) \to K(\mathbb Z^\wedge_p)\to K(\mathbb Q^\wedge_p)\). The conjecture also implies a long exact sequence of homotopy groups.
In the paper under reviewing the authors prove the conjecture for the case \(n=1\). In particular they prove the localization theorem and the main result of the paper is the dévissage theorem.

Reviewer: Do Ngoc Diep (Hanoi)

MSC:

18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
55N15	Topological \(K\)-theory

Keywords:

E-theory ring spectra; connective spectra; cofiber sequence

Citations:

Zbl 1019.18008; Zbl 0292.18004

References:

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