Advertisement

Chinese Annals of Mathematics, Series B

, Volume 40, Issue 1, pp 55–64 | Cite as

Triangulated Structures Induced by Triangle Functors

  • Zhibing Zhao
  • Xianneng Du
  • Yanhong BaoEmail author
Article
  • 15 Downloads

Abstract

Given a triangle functor F: \(\mathcal{A}\rightarrow\mathcal{B}\), the authors introduce the half image hImF, which is an additive category closely related to F. If F is full or faithful, then hImF admits a natural triangulated structure. However, in general, one can not expect that hImF has a natural triangulated structure. The aim of this paper is to prove that hImF admits a natural triangulated structure if and only if F satisfies the condition (SM). If this is the case, hImF is triangle-equivalent to the Verdier quotient \(\mathcal{A}\)/KerF.

Keywords

Triangulated category Triangle functor Half image Verdier quotient 

2000 MR Subject Classification

16E30 18A22 16E35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Happel, D., Triangulated Categories in the Representation Theory of Finite–Dimensional Algebras, Cambridge University Press, Cambridge, 1988.CrossRefzbMATHGoogle Scholar
  2. [2]
    Keller, B., Derived categories and universal problems, Comm. Algebra, 19(3), 1991, 699–747.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Neeman, A., Triangulated categories, Annals of Math. Studies, 148, Princeton University Press, Princeton, NJ, 2001.zbMATHGoogle Scholar
  4. [4]
    Rickard, J., Morita theory for derived categories, J. London Math. Soc., 39(2), 1989, 436–456.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Ringel, C. M. and Zhang, P., Objective tiangle functors, Sci. China Math., 58(2), 2014, 221–232.CrossRefGoogle Scholar
  6. [6]
    Ringel, C. M. and Zhang, P., From submodule categories to preprojective algebras, Math. Z., 278(1), 2014, 55–73.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Verider, J. L., Des, catégories dérivées abéliennes, Asterisque, 239, 1996, 111–125.(in French).Google Scholar
  8. [8]
    Zhang, P., Triangulated Categories and Derived Categories, Science Press, Beijing, 2015.Google Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiChina

Personalised recommendations