p-Adic Lifting Problems and Derived Equivalences

Abstract

For two derived equivalent $k$-algebras $\barΛ$ and $\barΓ$, we introduce a correspondence between $\OO$-orders reducing to $\barΛ$ and $\OO$-orders reducing to $\barΓ$. We outline how this may be used to transfer properties like uniqueness (or non-existence) of a lift between $\barΛ$ and $\barΓ$. 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 $\mathcal D^{κ,0}(2A)$ and $\mathcal D^{κ,0}(2B)$ can actually occur as basic algebras of blocks of group rings of finite groups.