Grothendieck groups in extriangulated categories
Grothendieck groups in extriangulated categories
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DOI：
10.1016/j.jalgebra.2021.01.029
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发表时间：
2019-12
影响因子：
0.9
通讯作者：
Bin Zhu;Zhuang Xiao
中科院分区：
文献类型：
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作者：
Bin Zhu;Zhuang Xiao
外三角范畴是由Nakaoka和Palu为了统一正合范畴和三角范畴的性质而引入的。本文考虑了外三角范畴C的Grothendieck群K 0（C）.证明了局部有限的外三角范畴C总是有几乎分裂扩张，并且在这种情况下，Grothendieck群K 0（C）的关系由Auslander-Rieten E-三角形生成.当限制于具有一个簇倾斜子范畴的三角范畴时，我们给出了一个部分匡威的结果：在具有一个簇倾斜子范畴的三角范畴中，Grothendieck群的关系由AR-三角形生成当且仅当三角范畴是局部有限的。我们还证明了包含G的象的K 0（C）的子群与C的稠密G-（co）分解子范畴之间存在一一对应，其中G是C的生成元，推广了关于三角范畴或正合范畴C的子范畴由K 0（C）的子群分类的结果.
Extriangulated categories were introduced by Nakaoka and Palu to give a unification of properties in exact categories and triangulated categories. We consider in this article the Grothendieck group K 0 (C) of an extriangulated category C. We show that a locally finite extriangulated category C always has almost split extensions and in this case the relations of the Grothendieck group K 0 (C) are generated by the Auslander-Rieten E-triangles. We give a partial converse result when restricting to the triangulated categories with a cluster tilting subcategory: in the triangulated category with a cluster tilting subcategory, the relations of the Grothendieck group are generated by AR-triangles if and only if the triangulated category is locally finite. We also show that there is a one-to-one correspondence between subgroups of K 0 (C) containing the image of G and dense G−(co) resolving subcategories of C where G a generator of C, which generalizes results about classifying subcategories of a triangulated or exact category C by subgroups of K 0 (C).