The Auslander-Reiten quiver of the category of m-periodic complexes

Abstract

Let $\mathcal{A}$ be an additive $k-$category and ${\mathbf{C}}_{\equiv m}(\mathcal{A})$ be the category of $m-$periodic objects. For any integer $m>1$, we study conditions under which the compression functor ${\mathcal{F}}_m:{\mathbf{C}}^b(\mathcal{A})\to {\mathbf{C}}_{\equiv m}(\mathcal{A})$ preserves or reflects irreducible morphisms. Moreover, we find sufficient conditions for the functor ${\mathcal{F}}_m$ to be a Galois $G$-covering in the sense of \cite{BL}. If in addition $\mathcal{A}$ is a dualizing category and $\mbox{mod}\, \mathcal{A}$ has finite global dimension then ${\mathbf{C}}_{\equiv m}(\mathcal{A})$ has almost split sequences. In particular, for a finite dimensional algebra $A$ with finite strong global dimension we determine how to build the Auslander-Reiten quiver of the category $\mathbf{C}_{\equiv m}(\mbox{proj}\, A)$. Furthermore, we study the behavior of sectional paths in $\mathbf{C}_{\equiv m}(\mbox{proj}\, A)$, whenever $A$ is any finite dimensional $k-$algebra over a field $k$.

24 pages