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

TL;DR

This paper establishes Morita and derived equivalences between blocks with generalized quaternion defect groups and three simple modules over large 2-adic rings, linking properties over the ring and residue field.

Contribution

It proves Morita-equivalence over the ring is equivalent to Morita-equivalence over the residue field for these blocks, and shows derived equivalence for blocks over the ring with the same defect group.

Findings

Morita-equivalence over the ring iff over the residue field

Derived equivalence for blocks with same defect group over the ring

Results apply to sufficiently large 2-adic rings

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.