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Let $(1)$ be an automorphism on an additive category $\mathcal{B}$, and let$\eta\colon (1)\to {\rm Id}_{\mathcal{B}}$ be a natural transformationsatisfying $\eta_{X(1)}=\eta_X(1)$ for any object $X$ in $\mathcal{B}$. Weconstruct a new Frobenius exact structure on the category of complexes in$\mathcal{B}$, which is associated to the natural transformation $\eta$. As aconsequence, the category introduced in Definition 2.4 of [J. Rickard, Moritatheory for derived categories, J. London Math. Soc. 39(1989), 436-456] has aFrobenius exact structure.

A new Frobenius exact structure on the category of complexes