Let A and B be two finite dimensional selfinjective K-algebra (K a field) that are both $\mathbb{Z}$-graded. Assume that the stable category of finitely generated $\mathbb{Z}$-graded A-modules is equivalent to the stable category of finitely generated $\mathbb{Z}$-graded B-modules. Is it true A is periodic if and only if B is periodic?
Here periodic for an algebra $A$ means that the bimodule $A$ is isomorphic to $\Omega^n(A)$ (syzygy taken in the category of bimodules)
