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DEGENERATING 0 IN TRIANGULATED CATEGORIES

Published online by Cambridge University Press:  08 June 2020

MANUEL SAORÍN
Affiliation:
Departemento de Matemáticas, Universidad de Murcia, Aptdo. 4021, 30100 Espinardo, Murcia, Spain email [email protected]
ALEXANDER ZIMMERMANN
Affiliation:
Université de Picardie, Département de Mathématiques et LAMFA (UMR 7352 du CNRS), 33 rue St Leu, F-80039 Amiens Cedex 1, France email [email protected]

Abstract

In previous work, based on the work of Zwara and Yoshino, we defined and studied degenerations of objects in triangulated categories analogous to the degeneration of modules. In triangulated categories ${\mathcal{T}}$, it is surprising that the zero object may degenerate. We show that the triangulated subcategory of ${\mathcal{T}}$ generated by the objects that are degenerations of zero coincides with the triangulated subcategory of ${\mathcal{T}}$ consisting of the objects with a vanishing image in the Grothendieck group $K_{0}({\mathcal{T}})$ of ${\mathcal{T}}$.

Type
Article
Copyright
© 2020 Foundation Nagoya Mathematical Journal

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Footnotes

The first named author was supported by research projects of 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.

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