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A new approach to Matsumoto’s theorem. (English) Zbl 0725.19001
The author gives a proof of Matsumoto’s theorem on \(K_ 2\) of a field using techniques from homological algebra. On considering a complex associated to the action of \(\mathrm{GL}(2,F)\) on \(\mathbb P^ 1(F)\) for a field \(F\) he derives the unstable presentation for \(H_ 0(F^{\bullet},H_ 2(\mathrm{SL}(2,F))\) and, on considering the action of \(\mathrm{GL}(n+1,F)\) on \(\mathbb P^ n(F)\) he proves the stability part of the theorem, that \(H_ 0(F^{\bullet},H_ 2(\mathrm{SL}(2,F))\) is isomorphic to \(H_ 2(\mathrm{SL}(F))=K_ 2(F)\).

MSC:
19C20 Symbols, presentations and stability of \(K_2\)
11R70 \(K\)-theory of global fields
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