期刊
BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY
卷 44, 期 4, 页码 2235-2252出版社
SPRINGERNATURE
DOI: 10.1007/s40840-020-01057-9
关键词
Extriangulated category; Proper class of E-triangles; Gorenstein homological dimensions
类别
资金
- NSF of China [11671069, 11771212]
- Qing Lan Project of Jiangsu Province
- Jiangsu Government Scholarship for Overseas Studies [JS-2019-328]
- National Natural Science Foundation of China [11901190, 11671221]
- Hunan Provincial Natural Science Foundation of China [2018JJ3205]
- Scientific Research Fund of Hunan Provincial Education Department [19B239]
This paper investigates the xi-Gprojective and xi-Ginjective dimensions of objects in an extriangulated category, providing characterizations of the xi-Gprojective dimension. The study shows that under certain assumptions, the equality between the two dimensions holds. These results extend the work in the exact category case and present a unique proof approach distinct from traditional methods in module or triangulated categories.
Let (C, E, s) be an extriangulated category with a proper class xi of E-triangles. In a previous work, we introduced and studied the xi-Gprojective and the xi-Ginjective dimension for any object in C. In this paper, we first give some characterizations of xi-Gprojective dimension by using derived functors on C. Consequently, we show that the following equality holds under some assumptions: sup{xi-GpdM | for any M is an element of C} = sup{xi-GidM | for any M is an element of C}, where xi-GpdM (resp., xi-GidM) denotes the xi-Gprojective dimension (resp., xi-Ginjective dimension) of M. As an application, our main results generalize the work by Bennis-Mahdou and Ren-Liu, which are newfor an exact category case. Moreover, our proof is far from the usual module or triangulated case.
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