Explicit description of boundary map in algebraic K-theory

Recall that for a DVR A with fraction field F and residue field k, there is a "localization" fiber sequence in algebraic K-theory,

$$K(k) \rightarrow K(A) \rightarrow K(F).$$

In Remark 5.17 of his "Higher Algebraic K-theory: I" paper, Quillen gives an explicit description of the corresponding boundary map $\partial:\Omega K(F) \rightarrow K(k)$, saying the proof will be in a later paper. My question is, has a proof appeared in the literature? I'd also be happy with proofs in the literature of any similar descriptions, e.g. involving the S-dot construction.

Thank you for reading!

edited Apr 8 2011 at 1:33

asked Apr 8 2011 at 0:05

Dustin Clausen
3,980●1●16●26

Have you looked at the similar post, for Milnor K-theory (which agrees with Quillen for $K_0$, $K_1$, $K_2$)? mathoverflow.net/questions/52177/… – Marty Apr 8 2011 at 0:54

I hadn't, thanks. But I do actually want the Quillen K-theory space-level description. – Dustin Clausen Apr 8 2011 at 0:57

I'm not exactly sure how how related it is, but Ross Staffeldt has a recent article that discusses the boundary maps arising from Waldhausen's fibration theorem. I believe the paper is in the Geom. Dedicata issue for Bruce Williams' birthday conference. – Dan Ramras Apr 8 2011 at 6:43

Very interesting question. it has been over one year and I was curious if the OP had found an answer. If so, I would request him to update! Thanks! – SGP Apr 22 2012 at 16:31

Apologies for not seeing this earlier: Could you please post it here? It would be very helpful. Thanks! – SGP Jan 24 at 0:37

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