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Suppose that $R\to S$ is a 1-connected morphism of connective structured ring spectra that induces an isomorphism on rational homotopy groups. Is the induced map of (Waldhausen) K-theory spectra $$ K(R) \to K(S) $$ also an equivalence on rational homotopy?

(The case of the map of group rings $S^0[G] \to \Bbb Z[G]$ was answered in the affirmative in one of Waldhausen's early papers.)

I am looking for a solid reference (assuming the result to be true; I believe it is).

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    $\begingroup$ For the record: The proof Waldhausen (Algebraic K-theory of topological spaces, I, Prop. 2.2, 1978) gave for S^0[G] --> Z[G] explicitly only uses that the map is a rational homotopy equivalence and 1-connected, hence also applies to any other such map of connective structured ring spectra. $\endgroup$
    – John Rognes
    Commented Jun 28, 2020 at 22:49

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The theorem can be found in more general form in Land, Tamme On the K-theory of pullbacks, Lemma 2.4.

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  • $\begingroup$ That is exactly what I am looking for! Thanks. $\endgroup$
    – John Klein
    Commented Jun 28, 2020 at 16:14
  • $\begingroup$ I saw the you mentioned that lemma in your talk today. Great talk by the way. I learned something... $\endgroup$
    – John Klein
    Commented Jun 29, 2020 at 18:03

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