Homological conjectures and stable equivalences of Morita type

Abstract

Let $ A $ and $ B $ be two finite-dimensional algebras over an algebraically closed field. Suppose that $ A $ and $ B $ are stably equivalent of Morita type; we prove that $ A $ satisfies the Auslander–Reiten conjecture (resp. Gorenstein projective conjecture, strong Nakayama conjecture, Auslander–Gorenstein conjecture, Nakayama conjecture, Gorenstein symmetric conjecture) if and only if $ B $ does so. This can provide new classes of algebras satisfying homological conjectures, and we give an example to illustrate it.

Keywords

• auslander–reiten conjecture
• gorenstein projective conjecture
• strong nakayama conjecture
• generalized nakayama conjecture
• stable equivalence of morita type
• gorenstein projective module