Covering theory, functor categories and the Krull-Gabriel dimension
Grzegorz Pastuszak
TL;DR
The paper develops a comprehensive framework linking covering theory for locally bounded K-categories, functor categories, and Krull-Gabriel dimension. It proves that a Galois G-covering $F:R\to A$ satisfies KG$(R)\le$KG$(A)$, using a tensor-product bifunctor to produce adjoints $\Phi,\Theta$ to the pull-up $\Psi$, and analyzes how these behave on finitely presented functors, leading to a Galois precovering structure for $\mathcal{F}(R)\to\mathcal{F}(A)$. In dense push-down scenarios the authors show $\Phi$ and $\Theta$ coincide on $\mathcal{F}(R)$ and remain exact and faithful, ensuring KG-dimension is preserved or controlled under coverings. The work introduces functors of the first and second kind, illustrates when $\Phi$ is dense with explicit examples, and derives concrete applications to Prest’s conjecture, including for algebras with strongly simply connected Galois coverings and weighted surface algebras where KG-dimension is finite or infinite as appropriate. Overall, the results provide a robust machinery to transfer representation-theoretic complexity via Galois coverings and to analyze how KG-dimension interacts with covering structures, with significant implications for domesticity and tame/wild classifications in the setting of finite-dimensional algebras.
Abstract
Assume that $K$ is an algebraically closed field, $R$ a locally bounded $K$-category, $G$ an admissible group of $K$-linear automorphisms of $R$ and $F:R\rightarrow A$ the Galois $G$-covering functor. In the first part of the paper we show that $KG(R)\leq KG(A)$ where $KG$ denotes the Krull-Gabriel dimension. This result is proved by developing the Galois covering theory of functor categories, based on the existence of the general tensor product bifunctor. We understand this theory as the theory of the left and the right adjoint functors $Φ,Θ:MOD(\mathcal{R})\rightarrow MOD(\mathcal{A})$ to the pull-up functor $Ψ=(F_λ)_{\bullet}:MOD(\mathcal{A})\rightarrow MOD(\mathcal{R})$, along the push-down functor $F_λ:\mathcal{R}\rightarrow\mathcal{A}$ where $\mathcal{R}=mod(R)$, $\mathcal{A}=mod(A)$ and $(F_λ)_{\bullet}=(-)\circ F_λ$. In the case $F_λ$ is dense, $Φ$ and $Θ$ are natural generalizations of the classical push-down functors. Generally, $Φ$ and $Θ$ restrict to categories $\mathcal{F}(R),\mathcal{F}(A)$ of finitely presented functors and the restricted functors coincide. In the second part of the paper, we show that $Φ:\mathcal{F}(R)\rightarrow\mathcal{F}(A)$ is a Galois $G$-precovering of functor categories. Next we consider an important special case when $R$ is simply connected locally representation-finite. Then $Φ$ may be studied in terms of classical covering theory which allows to give an example of $Φ$ which is not dense. This justifies the introduction of the functors of the first and the second kind, following the terminology of Dowbor and Skowroński. In the final part of the paper, we give special applications of our results. Last but not least, we discuss applications to the conjecture of M. Prest, relating the Krull-Gabriel dimension of an algebra with its representation type.