Czechoslovak Mathematical Journal, Vol. 72, No. 3, pp. 765-782, 2022
Stable tubes in extriangulated categories
Li Wang, Jiaqun Wei, Haicheng Zhang
Received April 17, 2021. Published online February 18, 2022.
Abstract: Let $\mathcal{X}$ be a semibrick in an extriangulated category. If $\mathcal{X}$ is a $\tau$-semibrick, then the Auslander-Reiten quiver $\Gamma(\mathcal{F}(\mathcal{X}))$ of the filtration subcategory $\mathcal{F}(\mathcal{X})$ generated by $\mathcal{X}$ is $\mathbb{Z}\mathbb{A}_{\infty}$. If $\mathcal{X}=\{X_i\}_{i=1}^t$ is a $\tau$-cycle semibrick, then $\Gamma(\mathcal{F}(\mathcal{X}))$ is $\mathbb{Z}\mathbb{A}_{\infty}/\tau_{\mathbb{A}}^t$.
References: [1] M. Gorsky, H. Nakaoka, Y. Palu: Positive and negative extensions in extriangulated categories. Available at https://arxiv.org/abs/2103.12482 (2021), 51 pages.
[2] O. Iyama, H. Nakaoka, Y. Palu: Auslander-Reiten theory in extriangulated categories. Available at https://arxiv.org/abs/1805.03776 (2019), 40 pages.
[3] H. Nakaoka, Y. Palu: Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah. Topol. Géom. Différ. Catég. 60 (2019), 117-193. MR 3931945 | Zbl 1451.18021
[4] C. M. Ringel: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics 1099. Springer, Berlin (1984). DOI 10.1007/BFb0072870 | MR 0774589 | Zbl 0546.16013
[5] D. Simson, A. Skowroński: Elements of the Representation Theory of Associative Algebras. Vol. 2. Tubes and Concealed Algebras of Euclidean Type. London Mathematical Society Student Texts 71. Cambridge University Press, Cambridge (2007). DOI 10.1017/CBO9780511619212 | MR 2360503 | Zbl 1129.16001
[6] L. Wang, J. Wei, H. Zhang: Semibricks in extriangulated categories. Commun. Algebra 49 (2021), 5247-5262. DOI 10.1080/00927872.2021.1940192 | MR 4328535 | Zbl 07431295
[7] P. Zhou, B. Zhu: Triangulated quotient categories revisited. J. Algebra 502 (2018), 196-232. DOI 10.1016/j.jalgebra.2018.01.031 | MR 3774890 | Zbl 1388.18014
Affiliations: Li Wang, School of Mathematics and Physics, Anhui Polytechnic University, Beijingzhong Road, Jiujiang District, Wuhu 241000, P. R. China, e-mail: wl04221995@163.com; Jiaqun Wei, Haicheng Zhang (corresponding author), Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, No.1 Wenyuan Road Qixia District, Nanjing 210023, P. R. China, e-mail: weijiaqun@njnu.edu.cn, zhanghc@njnu.edu.cn
