The liftability question for stable equivalences between representation-finite self-injective algebras

TL;DR

This paper proves that stable equivalences between all representation-finite self-injective algebras over an algebraically closed field lift to derived equivalences, resolving a long-standing open problem in algebra.

Contribution

It extends the liftability result to nonstandard cases and completes the proof for all representation-finite self-injective algebras, answering a question posed twenty years ago.

Findings

Stable equivalences lift to derived equivalences for all representation-finite self-injective algebras.

The proof fills a gap in the standard case, completing the theory.

The result confirms the Morita type nature of these equivalences.

Abstract

Let $k$ be an algebraically closed field. It is known that any stable equivalence between standard representation-finite self-injective $k$-algebras (without block of Loewy length 2) lifts to a standard derived equivalence, in particular, it is of Morita type. We show that the same holds for any stable equivalence between nonstandard representation-finite self-injective $k$-algebras. We also fill a gap in the original proof in standard case. This gives a complete solution of the liftability question raised by H. Asashiba about twenty years ago.