A Mayer-Vietoris sequence for Picard groups, with applications to integral group rings of dihedral and quaternion groups (Q1095224)

Let \(R\) be a Dedekind domain with field of fractions \(K\) and let \(\Lambda\) be an \(R\)-order in a separable \(K\)-algebra \(A\). For an \(R\)-subalgebra \(T\) of the center of \(\Lambda\), we denote by \(\text{Pic}_ T(\Lambda)\) the group of isomorphism classes \([M]\) of invertible \(\Lambda\)-bimodules centralized by \(T\). If \(e_ 1\) and \(e_ 2\) are central orthogonal idempotents of \(A\) with \(e_ 1+e_ 2=1\), we can express \(\Lambda\) as the fibre product \[ \begin{tikzcd}\Lambda\ar[r,"\mathrm{pr}_1"]\ar[d,"\mathrm{pr}_2" '] & \Lambda e_1=\Lambda_1\ar[d,"\varphi_2"]\\ \Lambda_2=\Lambda e_2\ar[r,"\varphi_2" '] & \Lambda\end{tikzcd} \] Let \(Pic_{e,T}(\Lambda)\) be the subgroup of \(\text{Pic}_ T(\Lambda)\) consisting of those \([M]\) where \(M\) is centralized by the \(e_ i\). If the kernel of each \(\varphi_ i\) is characteristic in the corresponding \(\Lambda_ i\), we obtain a Mayer-Vietoris sequence \[ 1\to u_ Z({\overline\Lambda})/\left<\varphi (u_ Z(\Lambda _ 1)),\varphi (u_ Z(\Lambda _ 1))\right>\to\text{Pic}_{e,T}(\Lambda)\to\text{Pic}_ T(\Lambda _ 1,\Lambda _ 2)\to 1, \] where \(\text{Pic}_ T(\Lambda _ 1,\Lambda _ 2)\) is a certain relative Picard group, and for a ring \(B\), \(u_ Z(B)\) is the group of units of the center of \(B\). We also obtain a somewhat similar sequence involving groups of outer automorphisms. Applications are given to the computation of Picard and outer automorphism groups for the integral group rings of finite groups of the following three types: (1) metacyclic groups of order \(pq\), where \(p\) is an odd prime and \(q\) divides \(p-1\); (2) dihedral 2-groups; and (3) generalized quaternion 2-groups.