Existence of Auslander-Reiten sequences in subcategories

Details Existence of Auslander-Reiten sequences in subcategories Existence of Auslander-Reiten sequences in subcategories

2009-11-03

arxiv.org/abs/0911.0633v1

Mathematics

Representation Theory

12 pages

Scientific paper

This paper studies the existence of Auslander-Reiten sequences in subcategories of mod A, where A is a finite dimensional algebra over a field. The two main theorems give necessary and sufficient conditions for the existence of Auslander-Reiten sequences in subcategories. Let M be a subcategory of mod A closed under extensions and direct summands, and let X be an indecomposable module in M such that Ext^{1}(X, X') is not zero for some X' in M. Then the following are equivalent: (i) DTrX has a precover in the stable category of mod A, (ii) There exists an Auslander-Reiten sequence 0 --> U --> W --> X --> 0 in M. We also have the dual result of the above theorem.

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