Blocks with a generalized quaternion defect group and three simple modules over a 2-adic ring

Abstract

We show that two blocks of generalized quaternion defect with three simple modules over a sufficiently large $2$-adic ring $\mathcal{O}$ are Morita-equivalent if and only if the corresponding blocks over the residue field of $\mathcal{O}$ are Morita-equivalent. As a corollary we show that any two blocks defined over $\mathcal{O}$ with three simple modules and the same generalized quaternion defect group are derived equivalent.