Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field \(\mathbb{F}_q \) and A be a finite dimensional k-algebra with a Frobenius morphism F. In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A ^F. As an application, it is shown that the derived category of A ^F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.