Covering theory, functor categories and the Krull-Gabriel dimension

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 $\Phi,\Theta:MOD(\mathcal{R})\rightarrow MOD(\mathcal{A})$ to the pull-up functor $\Psi=(F_{\lambda})_{\bullet}:MOD(\mathcal{A})\rightarrow MOD(\mathcal{R})$, along the push-down functor $F_{\lambda}:\mathcal{R}\rightarrow\mathcal{A}$ where $\mathcal{R}=mod(R)$, $\mathcal{A}=mod(A)$ and $(F_{\lambda})_{\bullet}=(-)\circ F_{\lambda}$. In the case $F_{\lambda}$ is dense, $\Phi$ and $\Theta$ are natural generalizations of the classical push-down functors. Generally, $\Phi$ and $\Theta$ 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 $\Phi:\mathcal{F}(R)\rightarrow\mathcal{F}(A)$ is a Galois $G$-precovering of functor categories. Next we consider an important special case when $R$ is simply connected locally representation-finite. Then $\Phi$ may be studied in terms of classical covering theory which allows to give an example of $\Phi$ which is not dense. This justifies the introduction of the functors of the first and the second kind, following the terminology of Dowbor and Skowro\'nski. 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.