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Matsumoto proved in his PhD thesis that if $F$ is a field then $$K_2(F)=(F^*\otimes F^*)/(x\otimes (1-x)).$$

The original Matsumoto proof as it is written in Milnor's book on algebraic K-theory looks not really nice to me and one guy told me that there is another proof of this fact that uses "sheaves of groups on Severi-Brauer varieties" which seems nicer to me. This guy told me that it is due to Vaserstein (Васерштейн) but it seems that Vaserstein was interested in other questions and couldn't give this proof. Perhaps you know whom this proof belongs to and I'd be very grateful if you could give me the reference to the article.

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Apparently, what this one guy meant is Merkurjev's proof of Merkurjev-Suslin theorem, which was writen down by A. R. Wadsworth and later by W. van der Kallen. The latter paper starts with the statement of Matsumoto theorem and the proof does indeed use cohomology of sheaves of K-groups on Severi-Brauer varieties.

As for the nicer proofs of Matsumoto theorem, one can take a look at A new approach to Matsumoto's theorem by K. Hutchinson or $(t^2−t)$-reciprocities on the affine line and Matsumoto's theorem by F. Keune.

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    $\begingroup$ Going to my Home Page to see that old paper of mine my impression is that Merkurjev uses Matsumoto's theorem. But notice that nowadays one also has the Milnor-Witt K-theory approach. $\endgroup$ – Wilberd van der Kallen Oct 15 '18 at 8:53

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