Auslander-Reiten translations in the monomorphism categories of exact categories

Abstract

Let $\Lambda$ be a finite dimensional algebra. Let $\mathcal C$ be a functorially finite exact subcategory of $\Lambda$-mod with enough projective and injective objects and $\mathcal S (\mathcal C)$ be its monomorphism category. It turns out that the category $\mathcal S (\mathcal C)$ has almost split sequences. We show an explicit formula for the Auslander-Reiten translation in $\mathcal S (\mathcal C)$. Furthermore, if $\mathcal C$ is a stably $d$-Calabi-Yau Frobenius category, we calculate objects under powers of Auslander-Reiten translation in the triangulated category $\overline{\mathcal S(\mathcal C)}$.