A new Frobenius exact structure on the category of complexes

Let (1) be an automorphism on an additive category BB, and let η:(1)→IdBη:(1)→IdB be a natural transformation satisfying ηX(1)=ηX(1)ηX(1)=ηX(1) for any object X   in BB. We construct a new Frobenius exact structure on the category of complexes in BB, which is associated to the natural transformation η. As a consequence, the category introduced in Definition 2.4 of Rickard (1989) [6] has a Frobenius exact structure.