Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
extriangulated category; semibrick; Auslander-Reiten quiver
extriangulated category; semibrick; Auslander-Reiten quiver
Summary:
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] Gorsky, M., Nakaoka, H., Palu, Y.: Positive and negative extensions in extriangulated categories. Available at https://arxiv.org/abs/2103.12482 (2021), 51 pages.
[2] Iyama, O., Nakaoka, H., Palu, Y.: Auslander-Reiten theory in extriangulated categories. Available at https://arxiv.org/abs/1805.03776 (2019), 40 pages.
[3] Nakaoka, H., Palu, Y.: Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah. Topol. Géom. Différ. Catég. 60 (2019), 117-193. MR 3931945 | Zbl 1451.18021
[4] Ringel, C. M.: Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics 1099. Springer, Berlin (1984). DOI 10.1007/BFb0072870 | MR 0774589 | Zbl 0546.16013
[5] Simson, D., Skowroński, A.: 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] Wang, L., Wei, J., Zhang, H.: Semibricks in extriangulated categories. Commun. Algebra 49 (2021), 5247-5262. DOI 10.1080/00927872.2021.1940192 | MR 4328535 | Zbl 07431295
[7] Zhou, P., Zhu, B.: Triangulated quotient categories revisited. J. Algebra 502 (2018), 196-232. DOI 10.1016/j.jalgebra.2018.01.031 | MR 3774890 | Zbl 1388.18014
Search
Partner of
