\(K_ 2\) of non-commutative local rings
A presentation for \(K_ 2\) of commutative local rings with finite residue fields using Dennis-Stein symbols is well-known [eg. see \textit{H. Maazen} and \textit{J. Stienstra}, J. Pure Appl. Algebra 10, 271-294 (1977; Zbl 0393.18013)]. This paper proves a noncommutative analog which is similar to the \textit{R. K. Dennis} and \textit{M. R. Stein} presentation for \(K_ 2\) of a discrete valuation ring [Adv. Math. 18, 182-238 (1975; Zbl 0318.13017)]. The author sets \(D_ 1(A)\) to be the group generated by Dennis-Stein symbols, i.e. \(<a,b>\) such that \(1+ab\in A^*\) (the group of units of A), and satisfying 4 relations. The relations are a little different from the usual commutative ones and are used to show that the two most complicated relations for a discrete valuation ring given by Dennis and Stein [loc. cit] are redundant. The main result in this paper is that \[ 0\to K_ 2(A)\to D_ 1(A)\to [A^*,A^*]\to 0 \] is exact for any local ring. If A is commutative, the sequence reduces to the usual presentation for \(K_ 2(A)\).
scientific article; zbMATH DE number 3944070
scientific article