On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras

Let \(K\) be an algebraically closed field, \(R\) a locally support-finite locally bounded \(K\)-category and \(G\) a torsion-free admissible group of \(K\)-linear automorphisms of \(R\). Recently Pastuszak showed that the induced Galois covering \(R\rightarrow R/ G\), where \(R/G\) denotes the orbit category, preserves the Krull-Gabriel dimension, i.e. \(\mathrm{KG}(R)=\mathrm{KG}(R/G)\). Therefore, in order to determine Krull-Gabriel dimensions of tame standard self-injective algebras it was sufficient to determine Krull-Gabriel dimensions of repetitive categories of tilted algebras of Dynkin type, tilted of Euclidean type or tubular algebras.

In this talk we recall the above results and show how they can be adapted to the case of cluster repetitive categories and cluster-tilted algebras which are their orbit categories. We will also give some background on the Galois coverings of functor categories, since it is the main tool used in these results, as well as present some related problems and applications. This is a report of a joint work with Grzegorz Pastuszak.