A new approach to Matsumoto's theorem (Q2277567)

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 \(H_ 2(\mathrm{SL}(F))=K_ 2(F)\).