A characterization of hereditary algebras via thick subcategories

Kimura, Yuta

doi:10.1090/proc/15036

A characterization of hereditary algebras via thick subcategories
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Proc. Amer. Math. Soc. 148 (2020), 2819-2822 Request permission

Abstract:

Let $\Lambda$ be a finite dimensional algebra. There exists a natural injective map from the set of wide subcategories in the category of finitely generated $\Lambda$-modules to the set of thick subcategories in the bounded derived category of $\Lambda$.We show, when $\Lambda$ is elementary, that the natural map is bijective if and only if $\Lambda$ is hereditary.

References

Takuma Aihara and Osamu Iyama, Silting mutation in triangulated categories, J. Lond. Math. Soc. (2) 85 (2012), no. 3, 633–668. MR 2927802, DOI 10.1112/jlms/jdr055
Kristian Brüning, Thick subcategories of the derived category of a hereditary algebra, Homology Homotopy Appl. 9 (2007), no. 2, 165–176. MR 2366948, DOI 10.4310/HHA.2007.v9.n2.a7
Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124, DOI 10.1017/CBO9780511629228
Kiyoshi Igusa, Shiping Liu, and Charles Paquette, A proof of the strong no loop conjecture, Adv. Math. 228 (2011), no. 5, 2731–2742. MR 2838056, DOI 10.1016/j.aim.2011.06.042
Henning Krause and Jan Šťovíček, The telescope conjecture for hereditary rings via Ext-orthogonal pairs, Adv. Math. 225 (2010), no. 5, 2341–2364. MR 2680168, DOI 10.1016/j.aim.2010.04.027
Chao Zhang and Hongyan Cai, A note on thick subcategories and wide subcategories, Homology Homotopy Appl. 19 (2017), no. 2, 131–139. MR 3714363, DOI 10.4310/HHA.2017.v19.n2.a8