The Auslander-Reiten duality via morphisms determined by objects. (English) Zbl 1386.18038
Let \(\mathcal{C}\) be a skeletally small \(k\)-linear Hom-finite Krull-Schmidt exact category. In a recent paper of P. Jiao [“The generalized Auslander-Reiten duality on an exact category”, arXiv:1609.07732], a generalized Auslander-Reiten duality was introduced. Let \(\mathcal{C}_\ell\) denote the smallest additive subcategory of \(\mathcal{C}\) containing the injectives and left terms of almost split conflations. The stable categories \(\underline{\mathcal{C}_r}\) and \(\overline{\mathcal{C}_\ell}\) can be defined, and the Auslander-Reiten transform gives an equivalence between these categories. The deflations which are right \(C\)-determined by an object \(C\) in the sense of Auslander are characterized in terms of \(\mathcal{C}_\ell\). Some related results are proved, including a criterion for existence of Auslander-Reiten duality.
Reviewer: Wolfgang Rump (Stuttgart)
MSC:
| 18E10 | Abelian categories, Grothendieck categories |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
References:
| [1] | Auslander, M., Functors and morphisms determined by objects, (Representation Theory of Algebras, Proc. Conf.. Representation Theory of Algebras, Proc. Conf., Temple Univ., Philadelphia, Pa., 1976. Representation Theory of Algebras, Proc. Conf.. Representation Theory of Algebras, Proc. Conf., Temple Univ., Philadelphia, Pa., 1976, Lect. Notes Pure Appl. Math., vol. 37 (1978), Dekker: Dekker New York), 1-244 · Zbl 0383.16015 |
| [2] | Auslander, M.; Reiten, I.; Smalø, S. O., Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, vol. 36 (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0834.16001 |
| [3] | Bian, N., Right minimal morphisms in arbitrary Krull-Schmidt categories, J. Math. (Wuhan), 29, 577-580 (2009) · Zbl 1212.18001 |
| [4] | Bühler, T., Exact categories, Expo. Math., 28, 1-69 (2010) · Zbl 1192.18007 |
| [5] | Chen, X.-W.; Le, J., A note on morphisms determined by objects, J. Algebra, 428, 138-148 (2015) · Zbl 1321.18001 |
| [6] | Jiao, P., The generalized Auslander-Reiten duality on an exact category (2016), preprint |
| [7] | Keller, B., Chain complexes and stable categories, Manuscr. Math., 67, 379-417 (1990) · Zbl 0753.18005 |
| [8] | Krause, H., Krull-Schmidt categories and projective covers, Expo. Math., 33, 535-549 (2015) · Zbl 1353.18011 |
| [9] | Krause, H., Morphisms determined by objects in triangulated categories, (Algebras, Quivers and Representations. Algebras, Quivers and Representations, Abel Symp., vol. 8 (2013), Springer: Springer Heidelberg), 195-207 · Zbl 1316.18014 |
| [10] | Lenzing, H.; Zuazua, R., Auslander-Reiten duality for abelian categories, Bol. Soc. Mat. Mexicana (3), 10, 169-177 (2004) · Zbl 1102.16011 |
| [11] | Ringel, C. M., Morphisms determined by objects: the case of modules over Artin algebras, Ill. J. Math., 56, 981-1000 (2012) · Zbl 1288.16012 |
| [12] | Ringel, C. M., The Auslander bijections: how morphisms are determined by modules, Bull. Math. Sci., 3, 409-484 (2013) · Zbl 1405.16030 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
