Triangulated quotient categories revisited
Triangulated quotient categories revisited
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重新审视三角商类别
DOI：
10.1016/j.jalgebra.2018.01.031
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发表时间：
2016-08
影响因子：
0.9
通讯作者：
Zhu Bin
中科院分区：
文献类型：
--
作者：
Zhou Panyue;Zhu Bin
中冈（Nakaoka）和帕鲁（Palu）通过提取精确类别和三角化类别之间的相似性来引入外三角化类别。定义了外三角范畴中子范畴的变异概念。设A是外三角范畴C的扩张闭子范畴。则加法商范畴M：= A/[X]在（A，A）形成X-突变对时自然带有三角结构。这个结果推广了三角范畴的许多同类结果。它被用来给出预三角化范畴C/[X]的厚三角化子范畴的一个分类，其中X在C中函有限。当C具有Auslander-Reiten平移τ时，我们证明了：对于C的函子有限子范畴X，若（C，C）是X-突变，则商C/[X]是三角范畴当且仅当（C，C）是X-突变，且当且仅当τ X _= X_∞.这推广了一个结果由Jørgensen谁证明了等价性之间的第一和第三条件的三角范畴。进一步证明了对于外三角范畴C的这样一个子范畴X，C有一个新的外三角结构，使得C是一个Frobenius外三角范畴。给出了精确范畴和三角范畴的应用。从这些应用中，我们提出了既不是正合范畴也不是三角范畴的外三角范畴。
Extriangulated categories were introduced by Nakaoka and Palu by extracting the similarities between exact categories and triangulated categories. A notion of mutation of subcategories in an extriangulated category is defined in this paper. Let A be an extension closed subcategory of an extriangulated category C. Then the additive quotient category M:= A/[X] carries naturally a triangulated structure whenever (A, A) forms an X-mutation pair. This result generalizes many results of the same type for triangulated categories. It is used to give a classification of thick triangulated subcategories of pre-triangulated category C/[X], where X is functorially finite in C. When C has Auslander–Reiten translation τ, we prove that for a functorially finite subcategory X of C containing projectives and injectives, the quotient C/[X] is a triangulated category if and only if (C, C) is X-mutation, and if and only if τ X _= X‾. This generalizes a result by Jørgensen who proved the equivalence between the first and the third conditions for triangulated categories. Furthermore, we show that for such a subcategory X of the extriangulated category C, C admits a new extriangulated structure such that C is a Frobenius extriangulated category. Applications to exact categories and triangulated categories are given. From the applications we present extriangulated categories which are neither exact categories nor triangulated categories.
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影响因子：
0.7
作者：
M. Auslander;Ø. Solberg
通讯作者：
M. Auslander;Ø. Solberg
DOI：
10.1073/pnas.1410635111
发表时间：
2014-06
期刊：
Proceedings of the National Academy of Sciences
影响因子：
--
作者：
B. Leclerc;L. Williams
通讯作者：
B. Leclerc;L. Williams
Frobenius 类别的三个结果
影响因子：
0.8
作者：
Chen, Xiao-Wu
通讯作者：
Chen, Xiao-Wu
影响因子：
0.8
作者：
M. Barot;D. Kussin;H. Lenzing
通讯作者：
M. Barot;D. Kussin;H. Lenzing
影响因子：
6.8
作者：
Theo Buehler;Theo B Uhler
通讯作者：
Theo Buehler;Theo B Uhler