Inductions and restrictions for stable equivalences of Morita type

Abstract

In this paper, we present two methods, induction and restriction procedures, to construct new stable equivalences of Morita type. Suppose that a stable equivalence of Morita type between two algebras $A$ and $B$ is defined by a $B$-$A$-bimodule $N$. Then, for any finite admissible set $\Phi$ and any generator $X$ of the $A$-module category, the $\Phi$-Auslander-Yoneda algebras of $X$ and $N\otimes_AX$ are stably equivalent of Morita type. Moreover, under certain conditions, we transfer stable equivalences of Morita type between $A$ and $B$ to ones between $eAe$ and $fBf$, where $e$ and $f$ are idempotent elements in $A$ and $B$, respectively. Consequently, for self-injective algebras $A$ and $B$ over a field without semisimple direct summands, and for any $A$-module $X$ and $B$-module $Y$, if the $\Phi$-Auslander-Yoneda algebras of $A\oplus X$ and $B\oplus Y$ are stably equivalent of Morita type for one finite admissible set $\Phi$, then so are the $\Psi$-Auslander-Yoneda algebras of $A\oplus X$ and $B\oplus Y$ for {\it every} finite admissible set $\Psi$. Moreover, two representation-finite algebras over a field without semisimple direct summands are stably equivalent of Morita type if and only if so are their Auslander algebras. As another consequence, we construct an infinite family of algebras of the same dimension and the same dominant dimension such that they are pairwise derived equivalent, but not stably equivalent of Morita type. This answers a question by Thorsten Holm.