Stable equivalences of Morita type for Φ‐Beilinson–Green algebras

The main focus of this paper is to present a method to construct new stable equivalences of Morita type. Suppose that a B‐A‐bimodule N define a stable equivalence of Morita type between finite dimensional algebras A and B. Then, for any generator X of the A‐module category and any finite admissible set Φ of natural numbers, the Φ‐Beilinson–Green algebras GAΦ(X) and GBΦ(N⊗AX) are stably equivalent of Morita type. In particular, if Φ={0}, we get a known result in literature. As another consequence, we construct an infinite family of derived equivalent algebras of the same dimension and of the same dominant dimension such that they are pairwise not stably equivalent of Morita type. Finally, we develop some techniques for proving that, if there is a graded stable equivalence of Morita type between graded algebras, then we can get a stable equivalence of Morita type between Beilinson–Green algebras associated with graded algebras.