Auslander-Reiten translations in the monomorphism categories of exact categories
Xiu-Hua Luo, Shijie Zhu
TL;DR
The paper develops a concrete AR-theory for monomorphism and epimorphism categories of exact categories, proving the existence of almost split sequences and giving explicit formulas for Auslander-Reiten translations. It achieves this by translating problems from ${\mathcal S}(\mathcal C)$ and ${\mathcal F}(\mathcal C)$ to the morphism and functor categories, using cw-exact frameworks, the RSS-equivalence, and stable/co-stable functors. A key contribution is the AR-translation formulas $\tau_{\mathcal S}(f) \cong {\mathbf Mimo}\tau_{\mathcal C}(\underline{\mathbf Cok}(f))$ and $\tau_{\mathcal F}(f) \cong {\mathbf Cok}\,{\mathbf Mimo}\tau_{\mathcal C}(\underline f)$, together with detailed transport mechanisms via the epivalence $\pi$ and the Theta functor. In the Frobenius and stably Calabi–Yau setting, the authors obtain powerful periodicity and suspension relations, e.g. $\tau_{\mathcal S}^6 f \cong f\langle 6d-4\rangle$, highlighting fractional Calabi–Yau-type behavior and enabling explicit computations in stable triangulated categories. The results extend classical AR theory from module categories to separated monomorphism/epimorphism contexts and yield concrete tools for studying Gorenstein/projective and Calabi–Yau phenomena in representation theory.
Abstract
Let $Λ$ be a finite dimensional algebra. Let $\mathcal C$ be a functorially finite exact subcategory of $Λ$-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 $\overline{\mathcal S(\mathcal C)}$.