Auslander-Reiten theory in extriangulated categories. (English) Zbl 1541.16017
Auslander-Reiten theory is a key tool to study the local structure of additive categories. The notion of an extriangulated category gives a unification of existing theories in exact or abelian categories and in triangulated categories. The authors develop Auslander-Reiten theory for extriangulated categories. This unifies Auslander-Reiten theories developed in exact categories and triangulated categories independently. The authors give two different sets of sufficient conditions on the extriangulated category so that existence of almost split extensions becomes equivalent to that of an Auslander-Reiten-Serre duality. They also show that existence of almost split extensions is preserved under taking relative extriangulated categories, ideal quotients, and extension-closed subcategories. Moreover, Iyama, Nakaoka, and Palu prove that the stable category of an extriangulated category is a \(\tau\)-category if it has enough projectives, almost split extensions and source morphisms. Finally the authors prove that any locally finite symmetrizable \(\tau\)-quiver is an Auslander-Reiten quiver of some extriangulated category with sink morphisms and source morphisms.
Reviewer: Mee Seong Im (Annapolis)
MSC:
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 18E10 | Abelian categories, Grothendieck categories |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 16G50 | Cohen-Macaulay modules in associative algebras |
Keywords:
Auslander-Reiten theory; extriangulated categories; exact categories; triangulated categories; Auslander-Reiten-Serre dualityReferences:
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