Auslander-Reiten theory in functorially finite resolving subcategories

Abstract

We analyze Auslander-Reiten quivers of functorially finite resolving subcategories. Chapter 1 gives a short introduction into the basic definitions and theorems of Auslander-Reiten theory in A-mod. We generalize these definitions and theorems in Chapter 2 and prove generalizations of the first and one and a half Brauer-Thrall conjecture for functorially finite resolving subcategories. Moreover, we show that sectional paths in Auslander-Reiten-quivers are invariants of decompositions of morphisms into sums of compositions of irreducible morphisms between indecomposable modules and are strongly connected to irreducible morphisms in subcategories. In Chapter 3 we introduce degrees of irreducible morphisms and use this notion to prove the generalization of the Happel-Preiser-Ringel theorem for functorially finite resolving subcategories. Finally, in Chapter 4, we analyze left stable components of Auslander-Reiten quivers and find out that their left subgraph types are given by Dynkin diagrams if and only if the corresponding subcategory is finite. In the preparation of the proof we discover connected components with certain properties and name them helical components due to their shape. It turns out later that these components are the same as coray tubes. In the final section we discuss under which conditions the length of modules tends to infinity if we knit to the left in a component and give a complete description of all connected components in which this is not the case.