Auslander-Reiten translations in the monomorphism categories of exact categories

TL;DR

This paper investigates the structure of monomorphism categories derived from exact categories over finite dimensional algebras, providing explicit formulas for Auslander-Reiten translations and exploring their behavior in Calabi-Yau contexts.

Contribution

It offers an explicit formula for Auslander-Reiten translation in monomorphism categories of exact categories and analyzes their properties in Calabi-Yau Frobenius categories.

Findings

Existence of almost split sequences in monomorphism categories.

Explicit formula for Auslander-Reiten translation in these categories.

Calculation of objects under powers of translation in Calabi-Yau settings.

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 $\overline{\mathcal S(\mathcal C)}$.