Cotorsion pairs, Gorenstein dimensions and triangle-equivalences
Cotorsion pairs, Gorenstein dimensions and triangle-equivalences
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发表时间：
2017-07
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通讯作者：
Jiangsheng Hu;Huanhuan Li;Jiaqun Wei;Xiaoyan Yang;Nanqing Ding
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作者：
Jiangsheng Hu;Huanhuan Li;Jiaqun Wei;Xiaoyan Yang;Nanqing Ding
设(A，B)是modR中的完全遗传余扭对。杨和丁在[56]中对复数的B维做了一般性的研究。本文利用(A，B)导出的模型结构，定义了复数的Gorenstein B维的概念，它可以用来描述复数的Gorenstein维对于任何完全遗传余扭对应该如何工作。给出了复数的Gorenstein B维的有限性的刻画。因此，我们研究了有限Gorenstein B维复的相对上同调群。此外，还给出了络合物的Gorenstein B维和B维之间的关系。其次，我们得到了遗传阿贝尔模型结构的同伦范畴、正则范畴的奇异性范畴和Forbenius范畴的稳定范畴之间的两个三角等价关系。作为应用，给出了初维猜想成立的一些充要条件。
Let (A, B) be a complete hereditary cotorsion pair in ModR. Yang and Ding made a general study of B dimensions of complexes in [56]. In this paper, we define the notion of Gorenstein B dimensions for complexes by applying the model structure induced by (A, B), which can be used to describe how Gorenstein dimensions of complexes should work for any complete hereditary cotorsion pair. Characterizations of the finiteness of Gorenstein B dimensions for complexes are given. As a consequence, we study relative cohomology groups for complexes with finite Gorenstein B dimensions. Moreover, the relationships between Gorenstein B dimensions and B dimensions for complexes are given. Next we get two triangle-equivalences between the homotopy category of a hereditary abelian model structure, the singularity category of an exact category and the stable category of a Forbenius category. As applications, some necessary and sufficient conditions for the validity of the Finitistic Dimension Conjecture are given.