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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
##### Keywords:
spectral sequence; Matsumoto’s theorem
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##### References:
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