Title: Quillen, Merkurjev and Suslin results about K2 of a conic
Tags: algebraic-k-theory
Question: Let $X$ be a conic without rational points over a field $F$ and $Q$ its associated quaternion algebra. The paper
https://www.math.ucla.edu/~merkurev/papers/residue.pdf
presents a proof of the exactness of the sequence
$$(1) \qquad  K_2(F) \to K_2(F(X)) \to \sum_x K_1(F(x))$$
where $F(X)$ is the field of functions of $X$ and the sum is over all points $x$ of the conic, $F(x)$ being the residual field at the point $x$.  This proof is elementary and involves computations in the quaternion algebra $Q$. The statement also is elementary, involving only $K_2$ of some fields.
As explained in this paper, the fact that (1) is exact was first discovered by Suslin 1982, by a more abstract argument using results of Quillen that
$$K_2(X) \to  K_2(F(X)) \to \sum_x K_1(F(x))$$
is exact, and the fact that $K_2(X)$ is the direct sum of $K_2(F)$ and $K_2(Q)$.
My question: is it possible to see this elementary proof as an &quot;unfolding&quot; of the abstract proof? Or to explain the connections between these two proofs?
Comment: Suslin's paper is "Quaternion homomorphism for the field of functions on a conic" and can be found online. Only beginning of page 2 is relevant here. It refers to a result of Dennis presented in the paper "On K 2 of finite dimensional division algebras over arithmetical fields" by Rehmann and Stuhler. By "unfolding" I meant something like you indicate.

Comment: I'm not qualified to answer, but could you give the publication data of the Suslin paper? It looks like an interesting question (I am supposing that "unfolding" means something like "beta reduction" in the informal sense https://ncatlab.org/nlab/show/beta-reduction#informal_usage). I hope the LaTeX didn't introduce any errors.


Here is another answer: No, the two proofs are quite different.

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