The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences
Let \(\Lambda\) be a tame hereditary Artin algebra and \({\mathcal F}=(\Lambda\)- mod\({}^{op},Ab)\) the category of all finitely presented contravariant functors from \(\Lambda\)-mod into the category of abelian groups. It is shown that \({\mathcal F}\) admits a filtration by Serre subcategories \(0={\mathcal F}_{-1}\subset {\mathcal F}_ 0\subset {\mathcal F}_ 1\subset {\mathcal F}_ 2={\mathcal F}\) where \({\mathcal F}_ i/{\mathcal F}_{i-1}\) is the full subcategory of \({\mathcal F}/{\mathcal F}_{i-1}\) consisting of all objects of finite length, \({\mathcal F}_ 1/{\mathcal F}_ 0\) is equivalent to the product category \({\mathcal R}\times {\mathcal R}\), for \({\mathcal R}\) the category of all regular \(\Lambda\)-modules, and \({\mathcal F}_ 2/{\mathcal F}_ 1\) to the category of all finite dimensional right vector spaces over the skew field \(End(_{\Lambda}Q)\), where \(_{\Lambda}Q\) denotes the unique indecomposable torsion-free divisible \(\Lambda\)-module. In particular \(\Lambda\)-mod has Krull-Gabriel dimension two and the above filtration leads to a hierarchy of exact sequences in \(\Lambda\)-mod. As an application, one obtains a description of morphisms between distinct Auslander-Reiten components of \(\Lambda\)-mod.
finitely presented contravariant functors
scientific article; zbMATH DE number 3953001
scientific article