Algebr Represent Theor https://doi.org/10.1007/s10468-018-9799-z Symmetry of the Definition of Degeneration in Triangulated Categories 1 2 Manuel Saor´ ın · Alexander Zimmermann Received: 16 December 2016 / Accepted: 10 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 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 degen- eration 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
Algebras and Representation Theory – Springer Journals
Published: May 28, 2018
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