Transfer of stable equivalences of Morita type

TL;DR

This paper investigates how stable equivalences of Morita type between finite-dimensional algebras can be transferred to their subalgebras defined by idempotents, with implications for representation-finite and n-representation-finite algebras.

Contribution

It establishes conditions under which stable equivalences of Morita type transfer between algebras and their idempotent subalgebras, extending to n-representation-finite algebras.

Findings

Stable equivalence of Morita type can be transferred to subalgebras via idempotents.

Auslander algebras of representation-finite algebras are stably equivalent if the original algebras are.

Extension of results to n-representation-finite and n-Auslander algebras.

Abstract

Let $A$ and $B$ be finite-dimensional $k$-algebras over a field $k$ such that $A/\rad(A)$ and $B/\rad(B)$ are separable. In this note, we consider how to transfer a stable equivalence of Morita type between $A$ and $B$ to that between $eAe$ and $fBf$, where $e$ and $f$ are idempotent elements in $A$ and in $B$, respectively. In particular, if the Auslander algebras of two representation-finite algebras $A$ and $B$ are stably equivalent of Morita type, then $A$ and $B$ themselves are stably equivalent of Morita type. Thus, combining a result with Liu and Xi, we see that two representation-finite algebras $A$ and $B$ over a perfect field are stably equivalent of Morita type if and only if their Auslander algebras are stably equivalent of Morita type. Moreover, since stable equivalence of Morita type preserves $n$-cluster tilting modules, we extend this result to $n$-representation-finite algebras and $n$-Auslander algebras studied by Iyama.