JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (8): 48-56.doi: 10.6040/j.issn.1671-9352.0.2022.558

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Gorenstein AC-representations of linear quivers

Qing SUN(),Gang YANG*()   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
  • Received:2022-10-31 Online:2023-08-20 Published:2023-07-28
  • Contact: Gang YANG E-mail:18265891818@163.com;yanggang@mail.lzjtu.cn

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Abstract:

Let Rep(Q, R) be the category of representations of R-modules over the linear quiver $ Q=(\bullet \rightarrow \bullet \rightarrow \cdots \rightarrow \bullet)$. It is mainly studied and characterized Gorenstein AC-injective representations and Gorenstein AC-flat representations in Rep(Q, R) in the paper.

Key words: quiver representations, Gorenstein AC-injective representations, Gorenstein AC-flat representations

CLC Number: 

  • O154.2
1 GABRIEL P . Unzelegbare darstellungen Ⅰ[J]. Manuscripta Mathematica, 1972, 6 (1): 71- 103.
doi: 10.1007/BF01298413
2 ENOCHS E , ESTRADA S . Projective representations of quivers[J]. Communications in Algebra, 2005, 33 (10): 3467- 3478.
doi: 10.1081/AGB-200058181
3 ENOCHS E , ESTRADA S , GARCÍAROZAS J R . Injective representations of infinite quivers. Applications[J]. Canadian Journal of Mathematics, 2009, 61 (2): 315- 335.
doi: 10.4153/CJM-2009-016-2
4 ENOCHS E , OYONARTE L , TORRECILLAS B . Flat covers and flat representations of quivers[J]. Communications in Algebra, 2004, 32 (4): 1319- 1338.
doi: 10.1081/AGB-120028784
5 ESHRAGHI H , HAFEZI R , SALARIAN S . Total acyclicity for complexes of representations of quivers[J]. Communications in Algebra, 2013, 41 (12): 4425- 4441.
doi: 10.1080/00927872.2012.701682
6 HOSSEINI E . Pure injective representations of quivers[J]. Bulletin of the Korean Mathematical Society, 2013, 50 (2): 389- 398.
doi: 10.4134/BKMS.2013.50.2.389
7 DALEZIOS G . Abelian model structures on categories of quiver representations[J]. Journal of Algebra and Its Applications, 2020, 19 (10): 2050195.
doi: 10.1142/S0219498820501959
8 DI Z , ESTRADA S , LIANG L , et al. Gorenstein flat representations of left rooted quivers[J]. Journal of Algebra, 2021, 584, 180- 214.
doi: 10.1016/j.jalgebra.2021.05.008
9 ODABAŞIS . Completeness of the induced cotorsion pairs in categories of quiver representations[J]. Journal of Pure and Applied Algebra, 2019, 223 (10): 4536- 4559.
doi: 10.1016/j.jpaa.2019.02.003
10 ENOCHS E E , JENDA O M G . Relative homological algebra: Volume 1[M]. Berlin: Walter de Gruyter, 2011.
11 STENSTRÖM B . Coherent rings and FP-injective modules[J]. Journal of the London Mathematical Society, 1970, (2): 323- 329.
12 HOLM H , JORGENSEN P . Cotorsion pairs in categories of quiver representations[J]. Kyoto Journal of Mathematics, 2019, 59 (3): 575- 606.
13 BRAVO D, GILLESPIE J, HOVEY M. The stable module category of a general ring[EB/OL]. 2014: arXiv: 1405.5768. https://arxiv.org/abs/1405.5768
14 王兴. Gorenstein AC-同调维数有限性的若干判别准则[D]. 兰州: 兰州交通大学, 2021.
WANG Xing. Some criteria for the finiteness of gorenstein AC-homology dimension[D]. Lanzhou: Lanzhou Jiaotong University, 2021.
15 孙情, 杨刚. 箭图表示的绝对Clean性质[J]. 西南师范大学学报(自然科学版), 2022, 47 (2): 16- 20.
SUN Qing , YANG Gang . Absolutely cleanness of quiver representations[J]. Journal of Southwest China Normal University (Natural Science Edition), 2022, 47 (2): 16- 20.
16 BRAVO D , ESTRADA S , IACOB A . FPn-injective, FPn-flat covers and preenvelopes, and Gorenstein AC-flat covers[J]. Algebra Colloquium, 2018, 25 (2): 319- 334.
doi: 10.1142/S1005386718000226
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