Abstract
We generalise Yoshino’s definition of a degeneration of two Cohen Macaulay modules to a definition of degeneration between two objects in a triangulated category. We derive some natural properties for the triangulated category and the degeneration under which the Yoshino-style degeneration is equivalent to the degeneration defined by a specific distinguished triangle analogous to Zwara’s characterisation of degeneration in module varieties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Riedtmann, C.: Degenerations for representations of quivers with relations. Annales de l’École Normale Supérieure 19, 275–301 (1986)
Zwara, G.: Degenerations of finite dimensional modules are given by extensions. Compositio Mathematica 121, 205–218 (2000)
Huisgen-Zimmermann, B., Saorín, M.: Geometry of chain complexes and outer automorphisms under derived equivalences. Trans. Am. Math. Soc. 353, 4757–4777 (2001)
Jensen, B.T., Su, X., Zimmermann, A.: Degenerations for derived categories. J. Pure Appl. Algebra 198, 281–295 (2005)
Jensen, B.T., Madsen, D., Su, X.: Degeneration of A-infinity modules. Trans. Am. Math. Soc. 361, 4125–4142 (2009)
Jensen, B.T., Su, X., Zimmermann, A.: Degeneration-like orders for triangulated categories. J. Algebra Appl. 4, 587–597 (2005)
Yoshino, Y.: Stable degeneration of Cohen-Macaulay modules. J. Algebra 332, 500–521 (2011)
Yoshino, Y.: On degeneration of Cohen-Macaulay modules. J. Algebra 248, 272–290 (2002)
Yoshino, Y.: On degenerations of modules. J. Algebra 278, 217–226 (2004)
Verdier, J.-L.: Des catégories dérivées des catégories abéliennes. Astérisque 239 (1996)
Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory. Springer, Heidelberg (1967)
May, P.: The axioms for triangulated categories, manuscript http://www.math.uchicago.edu/~may/MISCMaster.html
Zimmermann, A.: Representation theory: A homological algebra point of view. Springer, London (2014)
Keller, B.: On differential graded categories. In: Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006, vol. II, pp 151–190. European Mathematical Society (2006)
Keller, B.: Deriving DG categories. Annales de l’École Normale Supérieure 27, 63–102 (1994)
Zimmermann, A.: Remarks on a categorical definition of degeneration in triangulated categories. In: Proceedings of the 47th Symposion on Ring Theory and Representation Theory September 13-15, 2014. Saitama University, pp 138–145 (2015)
Chen, X.-W., Ye, Y., Zhang, P.: Algebras of derived dimension zero. Commun. Algebra 36, 1–10 (2008)
Matsumura, H.: Commutative ring theory. Cambridge University Press, Cambridge (1980)
Krause, H.: The stable category of a Noetherian scheme. Compositio Mathematica 141, 1128–1162 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Saorín is supported by research projects from the Secretaría de Estado de Investigación, Desarrollo e Innovación of the Spanish Government and the Fundación ‘Séneca’ of Murcia, with a part of FEDER funds. Zimmermann is supported by STIC Asie project ‘Escap’ of the Ministère des Affaires Étrangères de la France.
Rights and permissions
About this article
Cite this article
Saorín, M., Zimmermann, A. An Axiomatic Approach for Degenerations in Triangulated Categories. Appl Categor Struct 24, 385–405 (2016). https://doi.org/10.1007/s10485-015-9401-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10485-015-9401-3