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Symmetry of the Definition of Degeneration in Triangulated Categories

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Abstract

Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit. Riedtmann and Zwara gave an algebraic characterisation of this concept in terms of the existence of short exact sequences. Jensen, Su and Zimmermann, as well as independently Yoshino, studied the natural generalisation of the Riedtmann-Zwara degeneration to triangulated categories. The definition has an intrinsic non-symmetry. Suppose that we have a triangulated category in which idempotents split and either for which the endomorphism rings of all objects are artinian, or which is the category of compact objects in an algebraic compactly generated triangulated K-category. Then we show that the non-symmetry in the algebraic definition of the degeneration is inessential in the sense that the two possible choices which can be made in the definition lead to the same concept.

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Authors and Affiliations

  1. Departemento de Matemáticas, Universidad de Murcia, Aptdo. 4021, 30100, Espinardo, Murcia, Spain

    Manuel Saorín

  2. Université de Picardie, Département de Mathématiques et LAMFA (UMR 7352 du CNRS), 33 rue St Leu, 80039, Amiens Cedex 1, France

    Alexander Zimmermann

Authors
  1. Manuel Saorín
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  2. Alexander Zimmermann
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Correspondence to Alexander Zimmermann.

Additional information

Presented by: Kenneth Goodearl

The first named author is backed by research projects from the Ministerio de Economía y Competitividad of Spain (MTM2016-77445-P) and the Fundación ‘Séneca’ of Murcia (19880/GERM/15), both with a part of FEDER funds. He thanks these institutions for their support.

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Saorín, M., Zimmermann, A. Symmetry of the Definition of Degeneration in Triangulated Categories. Algebr Represent Theor 22, 801–836 (2019). https://doi.org/10.1007/s10468-018-9799-z

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