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数学进展 - 2018, Vol. 47(1): 155-159
研究论文
关于三角范畴中由对象决定的态射的一个注记
A Note on Morphisms Determined by Objects in Triangulated Categories

徐进德, 欧阳柏玉*
XU Jinde, OUYANG Baiyu

湖南师范大学数学与计算机科学学院, 长沙, 湖南, 410081
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan, 410081, P. R. China

出版日期: 2018-01-25
2018, Vol. 47(1): 155-159
DOI: 10.11845/sxjz.2016037b


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摘要 由对象决定的态射的概念由Auslander提出. 之后Krause证明了在含有Serre函子的Hom-有限的Krull-Schmidt $R$-线性三角范畴$\mathcal T$中, 任意态射$f$的决定子都存在.本文对该结论做了一个补充, 给出了$\mathcal T$ 中任意态射$f$的极小决定子$D(f)$的具体形式.
关键词 由对象决定的态射三角范畴    
Abstract:The notion of morphisms determined by an object was introduced by Auslander.Krause proved that a determiner for any morphism $f$ exists in a Hom-finite Krull-Schmidt $R$-linear triangulated category $\mathcal T$ having a Serre functor $S$.In this paper, we give the form of the minimal determiner $D(f)$ for any morphism $f$ in $\mathcal T$, which is a complement to the result of Krause.
Key wordsmorphisms determined by object    triangulated category
PACS:  O154.1  
基金资助:国家自然科学基金(No.11371131).
通讯作者: E-mail: $*$ oy@hunnu.edu.cn   
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[3] Auslander, M., Reiten, I. and Smalø, S.O., Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., Book 36,
Cambridge: Cambridge Univ. Press, 1997.
[4] Chen, X.-W. and Le, J., A note on morphisms determined by objects, J. Algebra, 2015, 428: 138-148.
[5] Krause, H., Morphisms determined by objects in triangulated categories, In: Algebras, Quivers and Representations (The Abel Symposium 2011) (Buan, A.B.,
Reiten, I. and Solberg, Øeds.), Abel Symp., Vol. 8, Berlin: Springer-Verlag, 2013, 195-207.
[6] Ringel, C.M., Morphisms determined by objects: the case of modules over Artin algebras, Illinois J. Math., 2012, 56(3): 981-1000.
[7] Ringel, C.M., The Auslander bijections: how morphisms are determined by modules, Bull. Math. Sci., 2013, 3(3): 409-484.
[1] 陈新红,卢明. 高维丛范畴中的丛倾斜对象[J]. 数学进展, 2016, 45(5): 641-651.
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