Higher Auslander-Reiten Theory
- Author(s): West, Jacob
- Advisor(s): Bergner, Julia E
- Gan, Wee Liang
- et al.
A development of Auslander-Reiten theory in the language of stable infinity-categories is presented. An infinity-category is a special kind of simplicial set which provides a common generalization of ordinary categories and nice topological spaces. Higher Auslander-Reiten theory can therefore be understood as a homotopy-theoretic analogue of the classical theory, which has proved to be an indispensable tool in many areas of representation theory.
We begin by introducing almost-split and irreducible morphisms in infinity-categories and establish their basic properties in direct analogy with the classical notions. We go on to describe morphisms determined by objects, a generalization of almost-split morphisms originally formulated by Auslander that until recently received little attention. We prove, using Brown representability, that every collection of maps with domain a compact object C uniquely determines, up to homotopy, a minimal C-determined morphism. From this result, the existence of almost-split and irreducible morphisms can be deduced.
We next describe the analogues of almost-split sequences in stable infinity-categories and prove that they exist in any compactly generated stable infinity-category with sufficiently small objects. Building on our earlier study of morphisms determined by objects, we then introduce Auslander functors on stable infinity-categories and show that they always exist on compactly generated stable infinity-categories. In good circumstances, Auslander functors specialize to Serre functors. This observation leads to a general duality formula which specializes to the classical Auslander-Reiten formula on the homotopy category.
Finally, we focus on an important class of examples of compactly generated stable infinity-categories associated to any Noetherian algebra over a complete local Noetherian ring. In this situation, we give a construction of an Auslander-Reiten translation functor and explain how it recovers the classical Auslander-Reiten translation on the associated (triangulated) homotopy category.