Normality of the elementary subgroups of twisted Chevalley groups over commutative rings
Let \(G\) be a twisted Chevalley group of type \({^2A_n}\) (\(n\geq 2\)), \({^2D_n}\) (\(n\geq 4\)) or \({^2E_6}\) over a commutative ring \(A\) with 1 and with an involutive automorphism \(\sigma\). Denote \(A_0=\{u\in A\mid u^\sigma=u\}\), \(U(A)=\{(a,b)\in A\times A\mid aa^\sigma=b+b^\sigma\}\) and \(U(A)^*=\{(a,b)\in U(A)\mid b\in A^*\}\), where \(A^*\) is the group of units of \(A\). Let \(E\) be the subgroup of \(G\) generated by root elements. It is proved that \(E\) is a normal subgroup of \(G\) if the ring \(A\) satisfies the following two hypotheses. (A1) For any maximal ideal \(m\) of \(A\), the natural map: \(A_0\to(A/(m\cap\sigma m))_0\) is surjective if \(G\) is not of type \({^2A_{2n}}\), and the natural map: \(U(A)\to U(A/(m\cap\sigma m))\) is surjective and \(U(A)^*\neq\emptyset\) if \(G\) is of type \({^2A_{2n}}\). (A2) For any maximal ideal \(m_0\) of \(A_0\), we have \(m_0=A_0\cap m_0A\). Note that if 2 is invertible in \(A\), then (A1) is satisfied.
scientific article; zbMATH DE number 783934