arXiv subjects
Publications and source records attributed to Shengyong Pan.
Let $A$ and $B$ be Gorenstein Artin algebras of Cohen-Macaulay finite type. We prove that, if $A$ and $B$ are derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived equivalent.
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 $Φ$ and any generator $X$ of the $A$-module category, the $Φ$-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 $Φ$-Auslander-Yoneda algebras of $A\oplus X$ and $B\oplus Y$ are stably equivalent of Morita type for one finite admissible set $Φ$, then so are the $Ψ$-Auslander-Yoneda algebras of $A\oplus X$ and $B\oplus Y$ for {\it every} finite admissible set $Ψ$. 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.
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.