Hermitian groups over local rings (Q2747687)

Let \(R\) be a ring with involution \(^{-}\) and let \(\lambda\) be a central element such that \(\lambda \bar{\lambda} = 1\). For any sequence \(a_1,\cdots,a_r\) of elements in \(R\), satisfying \(a_i = \lambda \bar{a_i}\), \( 1 \leq i\leq r\), and any \(n \geq r\) the \(n\)-th general Hermitian group \(GH_{2n}(R,a_1,\cdots,a_r)\) is the automorphism group of the metabolic form defined by the matrix NEWLINE\[NEWLINE\begin{bmatrix} A & \lambda I\\ I & 0 \end{bmatrix} ,NEWLINE\]NEWLINE where \(A = \text{diag} (a_1,\cdots, a_r, 0,\cdots ,0)\) [cf. \textit{G. Tang}, \(K\)-Theory 13, No. 3, 209-267 (1998; Zbl 0899.19003)].NEWLINENEWLINENEWLINEThe author shows that for a local ring the elementary Hermitian group \(EH_{2n}(R,a_1,\cdots,a_r)\) is a normal subgroup of \(GH_{2n}(R,a_1,\cdots,a_r)\) for \(n \geq r+2\) and is equal to the commutator subgroup of \(GH_{2n}(R,a_1,\cdots,a_r)\) if \(n \geq r+3\). Moreover, in the special case of a division ring he shows that the quotient group \(GH_{2n}(R,a_1,\cdots,a_r)/EH_{2n}(R,a_1,\cdots,a_r)\) is independent of the choice of \(a_1,\cdots,a_r\).