Transfer of stable equivalences of Morita type

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.