TITLE: Quillen, Merkurjev and Suslin results about K2 of a conic
QUESTION [9 upvotes]: 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 "unfolding" of the abstract proof? Or to explain the connections between these two proofs?

REPLY [11 votes]: No, the two proofs are quite different.