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


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.


Triangulated category Triangle functor Half image Verdier quotient 

2000 MR Subject Classification

16E30 18A22 16E35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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