On covering theory and its applications
Grzegorz Pastuszak
TL;DR
The paper surveys covering theory for quivers and locally bounded $K$-categories and extends it to module and functor categories via Galois coverings, establishing a framework that preserves and controls Krull-Gabriel dimension under coverings. It develops push-down and pull-up adjunctions, and introduces functors of the first and second kind to classify indecomposable functors across coverings, highlighting when the functor $oldsymbol{ m extPhi}$ is dense. The results are then applied to key algebra classes, including repetitive algebras, cluster-tilted algebras, and algebras with strongly simply connected Galois coverings, yielding precise KG-dimension characterizations (0, 2, or $ inite$) tied to Dynkin, Euclidean, tubular, domestic, and polynomial-growth types. These insights support Prest’s domestic-type conjecture in broad families and illustrate how covering theory informs the structure of finitely presented functors and the Krull-Gabriel filtration. Overall, the work integrates covering theory with the KG-dimension program to provide concrete criteria for finiteness and domesticity across important representation-theoretic settings.
Abstract
The aim of this survey is to present applications of covering techniques in the theory of Krull-Gabriel dimension. We start with recalling fundamental facts of the classical covering theory of quivers and locally bounded categories. Then we present some recent results on covering theory of functor categories. These are interesting themselves, but also allow to relate Krull-Gabriel dimensions of locally bounded categories $R$ and $A$ when $R\rightarrow A$ is a Galois covering functor. Finally, we concentrate on applications of our methods in describing Krull-Gabriel dimensions of various classes of algebras and locally bounded categories.