p-Adic Lifting Problems and Derived Equivalences

TL;DR

This paper explores how derived equivalences between algebras can be used to transfer lifting properties of orders over a discrete valuation ring, with applications to classifying certain block algebras of finite groups.

Contribution

It introduces a correspondence between orders reducing to derived equivalent algebras and demonstrates its use in transferring lift properties, with applications to group ring blocks.

Findings

Only specific tame dihedral algebras occur as basic blocks of finite group rings.

The correspondence helps determine lift uniqueness and existence.

Application to classifying block algebras of finite groups.

Abstract

For two derived equivalent $k$-algebras $\bar\Lambda$ and $\bar\Gamma$, we introduce a correspondence between $\OO$-orders reducing to $\bar\Lambda$ and $\OO$-orders reducing to $\bar\Gamma$. We outline how this may be used to transfer properties like uniqueness (or non-existence) of a lift between $\bar\Lambda$ and $\bar\Gamma$. As an application, we look at tame algebras of dihedral type with two simple modules, where, most notably, we are able to show that among those algebras only the algebras $\mathcal D^{\kappa,0}(2A)$ and $\mathcal D^{\kappa,0}(2B)$ can actually occur as basic algebras of blocks of group rings of finite groups.