On Picard groups of blocks of finite groups

Abstract

We show that the subgroup of the Picard group of a $p$-block of a finite group given by bimodules with endopermutation sources modulo the automorphism group of a source algebra is determined locally in terms of the fusion system on a defect group. We show that the Picard group of a block over the a complete discrete valuation ring $\mathcal{O}$ of characteristic zero with an algebraic closure $k$ of ${\mathbb{F}}_p$ as residue field is a colimit of finite Picard groups of blocks over $p$-adic subrings of $\mathcal{O}$. We apply the results to blocks with an abelian defect group and Frobenius inertial quotient, and specialise this further to blocks with cyclic or Klein four defect groups.