Dimension theory and nonstable \(K_{1}\) of quadratic modules
For an integer \(n\geq 3\) and a ring \(A\) with finite Bass--Serre dimension \(d(A)\), \textit{A.~Bak} [\(K\)-Theory 4, 363-397 (1991; Zbl 0741.19001)] proved that the non-stable \(K\)-group \(K_{1,n}(A):= GL_n(A)/E_n(A)\) is a nilpotent-by-abelian group. The aim of this paper is to prove the quadratic analog of Bak's result. Let \((A,\Lambda)\) be a form ring. Let \(G_{2n}(A,\Lambda)\) denote the general quadratic group of rank \(n\) over \((A,\Lambda)\), and \(E_{2n}(A,\Lambda)\) its elementary subgroup. The author constructs a dimension filtration \(G_{2n}(A,\Lambda)\supseteq G_{2n}^0(A,\Lambda)\supseteq G_{2n}^1(A,\Lambda)\supseteq \cdots\supseteq E_{2n}(A,\Lambda)\) of the general quadratic group \(G_{2n}(A,\Lambda)\) such that \(G_{2n}(A,\Lambda)/G_{2n}^0(A,\Lambda)\) is abelian, \(G_{2n}^0(A,\Lambda)\supseteq G_{2n}^1(A,\Lambda)\supseteq \cdots\) is a descending central series, and \(G_{2n}^{d(A)}(A,\Lambda) =E_{2n}(A,\Lambda)\). In particular, \(K_{1,2n}(A,\Lambda) :=G_{2n}(A,\Lambda)/E_{2n}(A,\Lambda)\) is a nilpotent-by-abelian group (thus solvable) when \(d(A)< \infty\).
scientific article; zbMATH DE number 1925075
scientific article