Covering techniques in higher Auslander-Reiten theory

TL;DR

This paper explores how higher Auslander-Reiten theory concepts, especially $n$-precluster tilting subcategories, behave under Galois coverings, establishing preservation results and applications to $ au_n$-tilting theory.

Contribution

It demonstrates that $n$-precluster tilting subcategories and $ au_n$-tilting finiteness are preserved under Galois coverings in higher Auslander-Reiten theory.

Findings

Push-down functor maps $G$-equivariant $n$-precluster tilting subcategories to similar subcategories.

Equivalence of $n$-minimal Auslander-Gorenstein property under Galois coverings.

Support $ au_n$-tilting finiteness is preserved under Galois coverings.

Abstract

This paper investigates the behavior of $n$-precluster tilting subcategories under the push-down functor in the context of Galois coverings of locally bounded categories. Building on higher Auslander-Reiten theory and covering techniques, we establish that for a locally support-finite category $\mathcal{C}$ with a free group action $G$ on its indecomposables, the push-down functor maps $G$-equivariant $n$-precluster tilting subcategories of ${\rm mod}\mbox{-}\mathcal{C}$ to $n$-precluster tilting subcategories of ${\rm mod}\mbox{-}(\mathcal{C}/G)$, and vice versa. These results provide a framework for studying $\tau_n$-selfinjective algebras. We further prove that ${\rm mod}\mbox{-}\mathcal{C}$ is $n$-minimal Auslander-Gorenstein if and only if ${\rm mod}\mbox{-}(\mathcal{C}/G)$ is so, under square-free conditions on $\mathcal{C}/G$. Additionally, we analyze support $\tau_n$-tilting pairs via the push-down functor, showing that locally $\tau_n$-tilting finiteness is preserved under Galois coverings. Our work offers new insights into the interplay between higher homological algebra and covering theory in representation-finite contexts.