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The Extension Dimension of Abelian Categories

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Abstract

Let \(\mathcal {A}\) be an abelian category having enough projective objects and enough injective objects. We prove that if \(\mathcal {A}\) admits an additive generating object, then the extension dimension and the weak resolution dimension of \(\mathcal {A}\) are identical, and they are at most the representation dimension of \(\mathcal {A}\) minus two. By using it, for a right Morita ring Λ, we establish the relation between the extension dimension of the category mod Λ of finitely generated right Λ-modules and the representation dimension as well as the right global dimension of Λ. In particular, we give an upper bound for the extension dimension of mod Λ in terms of the projective dimension of certain class of simple right Λ-modules and the radical layer length of Λ. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements.

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Acknowledgements

This work was partially supported by NSFC (No. 11571164), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, Postgraduate Research and Practice Innovation Program of Jiangsu Province (Grant No. KYCX17_0019). The authors thank the referee for very useful and detailed suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, Jiangsu Province, People’s Republic of China

    Junling Zheng & Zhaoyong Huang

  2. College of Science, Henan University of Engineering, Zhengzhou, 451191, Henan Province, People’s Republic of China

    Xin Ma

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  1. Junling Zheng
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  2. Xin Ma
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  3. Zhaoyong Huang
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Correspondence to Zhaoyong Huang.

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Zheng, J., Ma, X. & Huang, Z. The Extension Dimension of Abelian Categories. Algebr Represent Theor 23, 693–713 (2020). https://doi.org/10.1007/s10468-019-09861-z

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