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Borel famously used analysis on symmetric spaces to compute the rationalised algebraic $K$-theory groups of rings of integers $\mathcal{O}_F$ in number fields, e.g. $K_i(\mathbb{Z}) \otimes \mathbb{Q}$ vanishes unless $i=5,9,13,\ldots$ in which case it is $\mathbb{Q}$. Nowadays we can also access the algebraic $K$-theory groups of $\mathcal{O}_F$ using the Quillen-Lichtenbaum conjecture, and I am wondering whether this might give an analysis-free computation of these algebraic $K$-theory groups.

Does the proof of the Quillen-Lichtenbaum conjecture require Borel's computation as input?

If it does not, the following question makes sense:

Can we use Quillen-Lichtenbaum to recover Borel's computation? If not, what can it prove about the algebraic $K$-theory groups of $\mathcal{O}_F$ if we are not allowed to use Borel's computation?

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  • $\begingroup$ The Quillen-Lichtenbaum Conjecture is a conjecture. Which proof are you referring to? $\endgroup$
    – Alex B.
    Commented Nov 10, 2021 at 18:09
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    $\begingroup$ @AlexB does not it follow from Bloch-Kato conjecture, which is proved? $\endgroup$
    – Fedor Petrov
    Commented Nov 10, 2021 at 18:19
  • $\begingroup$ I am very sorry, I shouldn't comment when I am this tired. I was thinking of the Lichtenbaum conjecture on special values of zeta functions in terms of K-groups. $\endgroup$
    – Alex B.
    Commented Nov 10, 2021 at 18:32
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    $\begingroup$ A citation from the K-theory Handbook (see the link in the question), page 359. " If one had finite generation results for motivic cohomology, Conjecture 1 would follow from all this work." Finite generation is a difficult question. $\endgroup$
    – Mikhail Bondarko
    Commented Nov 13, 2021 at 16:47

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