Abstract
The concept of a morphism determined by an object provides a method to construct or classify morphisms in a fixed category. We show that this works particularly well for triangulated categories having Serre duality. Another application of this concept arises from a reformulation of Freyd’s generating hypothesis.
Keywords
- Morphisms Ending
- Auslander-Reiten Formula
- Stable Module Category
- Sphere Spectrum
- Full Additive Subcategory
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- 1.
This result is not correct as stated; the term \(P(\operatorname{Coker} \alpha )\) needs to be modified, as pointed out by Ringel in [14].
References
M. Auslander, Functors and morphisms determined by objects, Representation theory of algebras (Proc. conf., Temple Univ., Philadelphia, Pa., 1976), 1–244, Lecture Notes in Pure Appl. Math. 37, Dekker, New York, 1978.
M. Auslander, Applications of morphisms determined by modules, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976), 245–327, Lecture Notes in Pure Appl. Math., 37, Dekker, New York, 1978.
M. Auslander, I. Reiten, Stable equivalence of dualizing R-varieties, Adv. Math. 12 (1974), 306–366.
M. Auslander, I. Reiten, Representation theory of Artin algebras III. Almost split sequences, Commun. Algebra 3 (1975), 239–294.
M. Auslander, I. Reiten, S. O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge Univ. Press, Cambridge, 1995.
A. Beligiannis, Auslander-Reiten triangles, Ziegler spectra and Gorenstein rings, K-Theory 32, no. 1 (2004), 1–82.
P. Freyd, Stable homotopy, in Proc. conf. categorical algebra (La Jolla, Calif., 1965), 121–172, Springer, New York, 1966.
O. Iyama, Y. Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172, no. 1 (2008), 117–168.
H. Krause, Decomposing thick subcategories of the stable module category, Math. Ann. 313, no. 1 (1999), 95–108.
H. Krause, Smashing subcategories and the telescope conjecture—an algebraic approach, Invent. Math. 139, no. 1 (2000), 99–133.
H. Krause, Auslander-Reiten theory via Brown representability, K-Theory 20, no. 4 (2000), 331–344.
H. Krause, Report on locally finite triangulated categories, K-Theory 9, no. 3 (2012), 421–458.
I. Reiten, M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J. Am. Math. Soc. 15, no. 2 (2002), 295–366.
C. M. Ringel, Morphisms determined by objects: the case of modules over Artin algebras, Ill. J. Math., to appear. arXiv:1110.6734.
Acknowledgements
Some 20 years ago, Maurice Auslander encouraged me (then a postdoc at Brandeis University) to read his Philadelphia notes [1], commenting that they had never really been used. More recently, postdocs at Bielefeld asked me to explain this material; I am grateful to both of them. Special thanks goes to Greg Stevenson for helpful discussions and comments on a preliminary version of this paper, and to Apostolos Beligiannis for sharing interest in this subject.
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Krause, H. (2013). Morphisms Determined by Objects in Triangulated Categories. In: Buan, A., Reiten, I., Solberg, Ø. (eds) Algebras, Quivers and Representations. Abel Symposia, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39485-0_9
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