2009/06/09 by Shengyong Pan, Pan, Shengyong, Changchang Xi +1
Mathematics · #FOS: Mathematics #Representation Theory (math.RT) #math.RT
paper · pdf · doi:10.48550/arxiv.0906.1647
arxiv created 2009/12/02 · arxiv updated 2009/12/08
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.