Almost split sequences in tri-exact categories. (English) Zbl 1511.16004
The authors unify and extend some existence theorems for almost split sequences in abelian categories and exact categories, by working with tri-exact categories. First, they show that every exact category is equivalent to a tri-exact category with equivalent stable categories. This ensures that the study of almost split sequences in exact categories and abelian categories is covered under the tri-exact setting. Then, it is possible to obtain necessary and sufficient conditions for the existence of an almost split sequence in a tri-exact category; and to show that a tri-exact R-category has almost split sequences on the right (or left) if and only if it admits a full right (or left) Auslander-Reiten functor.
Finally, if an abelian category admits a Nakayama functor with respect to a subcategory of projective objects they establish the existence of an almost split triangle in the bounded derived category.
Finally, if an abelian category admits a Nakayama functor with respect to a subcategory of projective objects they establish the existence of an almost split triangle in the bounded derived category.
Reviewer: Luz Adriana Mejia Castaño (Barranquilla)
MSC:
| 16D90 | Module categories in associative algebras |
| 16G20 | Representations of quivers and partially ordered sets |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 16E35 | Derived categories and associative algebras |
Keywords:
algebras; almost split sequences; almost split triangles; abelian categories; derived categories; triangulated categoriesReferences:
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