Auslander's defect formula and a commutative triangle in an exact category

Abstract

We prove Auslander's defect formula in an exact category, and obtain a commutative triangle involving the Auslander bijections and the generalized Auslander{Reiten duality.

Keywords

Auslander's defect formula / Auslander bijections / morphisms determined by objects

References

[1] Auslander M. Functors and morphisms determined by objects. In: Gordon R, ed. Representation Theory of Algebras: Proceedings of the Philadelphia Conference. Lecture Notes in Pure and Applied Mathematics, Vol 37. New York: Marcel Dekker, 1978, 1–244

[2] Auslander M, Reiten I, Smal S O. Representation Theory of Artin Algebras. Cambridge Stud Adv Math, Vol 36. Cambridge: Cambridge Univ Press, 1995

[3] Chen X W. The Auslander bijections and universal extensions. Ark Mat, 2017, 55: 41–59

[4] Iyama O. Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. Adv Math, 2007, 210: 22–50

[5] Jasso G, Kvamme S. An introduction to higher Auslander-Reiten theory. Bull Lond Math Soc, 2019, 51: 1–24

[6] Jiao P. The generalized Auslander-Reiten duality on an exact category. J Algebra Appl, 2018, 17: 1850227

[7] Jiao P, Le J. The Auslander-Reiten duality via morphisms determined by objects. J Pure Appl Algebra, 2018, 222: 807–817

[8] Keller B. Chain complexes and stable categories. Manuscripta Math, 1990, 67: 379–417

[9] Krause H. A short proof for Auslander's defect formula. Linear Algebra Appl, 2003, 365: 267–270

[10] Lenzing H, Zuazua R. Auslander-Reiten duality for abelian categories. Bol Soc Mat Mex (3), 2004, 10: 169–177

[11] Ringel C M. Morphisms determined by objects: the case of modules over Artin algebras. Illinois J Math, 2012, 56: 981–1000

[12] Ringel C M. The Auslander bijections: how morphisms are determined by modules. Bull Math Sci, 2013, 3: 409–484

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature