n-角范畴和n-阿贝尔范畴的一些注记
Some Notes on n-Angulated Categories andn-Abelian Categories
DOI: 10.12677/PM.2025.159242, PDF,   
作者: 周高雄：云南师范大学数学学院，云南 昆明
关键词: n-角范畴；n-阿贝尔范畴；半单范畴；n-Angulated Categories； n-Abelian Categories； Semisimple Categories
摘要: 本文主要研究n-角范畴与n-阿贝尔范畴的相关性质,证明了以下结论:若C是n-角范畴则C是半单范畴;若C 是n-阿贝尔范畴,且是半单范畴,则C为阿贝尔范畴,且对C的任意自同构∑:C →C,(C,Σ)均可构成n-角范畴,还探讨了半单阿贝尔范畴中(n+1)-序列同构于平凡n-序列有限直和的条件，为高阶同调代数中范畴结构研究提供新依据。
Abstract: In this paper, we focus on some properties related on n-angulated categories and n-Abelian categories, and we prove the following results: if C is a n-angulated categories then C is semisimple; if C is a n-Abelian categories which is semisimple, then C is an Abel category, and for any self-isomorphism ∑:C →C,(C,Σ) is a n-angulated categories. Moreover, we establish necessary and sufficient condition for a (n + 1)- sequence isomorphic to a finite direct sums of ordinary n-sequences, which provide a new horizon for the study of high-dimension homological algebra.
文章引用：周高雄. n-角范畴和n-阿贝尔范畴的一些注记[J]. 理论数学, 2025, 15(9): 146-164. https://doi.org/10.12677/PM.2025.159242

参考文献

[1] Grothendieck, A. (1957) Sur quelques points d'algebre homologique, I. Tohoku Mathematical Journal, 9, 119-221.
https://doi.org/10.2748/tmj/1178244839
[2] Verdier, J.L. (1997)Des catégories dérivées abeliennes. Astérisque (1996), No. 239, xii + 253pp. With a preface by L. lllusie. Edited and with a note by G. Maltsiniotis.
[3] Kontsevich, M. and Soibelman, Y. (2011)Cohomological Hall Algebra, Exponential HodgeStructures and Motivic Donaldson-Thomas Invariants. Communications in Number Theoryand Physics, 5, 231-252.
https://doi.org/10.4310/cntp.2011.v5.n2.a1
[4] Kontsevich, M. and Soibelman, Y.(2008)Stability Structures, Motivic Donaldson-ThomasInvariants and Cluster Transformations. arXiv: 0811.2435
[5] lyama, 0.(2011)Cluster Tilting for higher Auslander algebras. Advances in Mathematics,226,1-61.
https://doi.org/10.1016/j.aim.2010.03.004
[6] lyama, 0. (2007) Higher-Dimensional Auslander-Reiten Theory on Maximal Orthogonal Sub.categories. Advances in Mathematics, 210, 22-50.
https://doi.org/10.1016/j.aim.2006.06.002
[7] Geiss, C., Keller, B. and Oppermann, S. (2013) n-Angulated Categories. Journal fur die reine und angewandte Mathematik (Crelles Journal), 2013, 101-120.
https://doi.org/10.1515/crelle.2011.177
[8] Jasso, G. (2016)N-Abelian and N-Exact Categories. Mathematische Zeitschrift, 283, 703-759
https://doi.org/10.1007/s00209-016-1619-8
[9] Luo, D.(2017)Homological Algebra in n-Abelian Categories. Proceedings-Mathematical Sciences,127,625-656.
https://doi.org/10.1007/s12044-017-0345-4
[10] 章璞.三角范畴与导出范畴[M].北京: 科学出版社,2015.
[11] Kalck, M. (2021)A New Equivalence between Singularity Categories of Commutative AlgebrasAdvances in Mathematics, 390, Article 107913.
https://doi.org/10.1016/j.aim.2021.107913

为你推荐