Homological conjectures and stable equivalences of Morita type

Juxiang Sun; Guoqiang Zhao; Juxiang Sun; Guoqiang Zhao

doi:10.3934/math.2025120

AIMS Mathematics

2025, Volume 10, Issue 2: 2589-2601. doi: 10.3934/math.2025120

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Homological conjectures and stable equivalences of Morita type

Juxiang Sun ^{1
,
,},
Guoqiang Zhao ²

1.
School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu 476000, China
2.
School of Science, Hangzhou Dianzi University, Hangzhou 310018, China

Received: 12 December 2024 Revised: 16 January 2025 Accepted: 23 January 2025 Published: 13 February 2025
MSC : 16E10, 16E30

Let $ A $ and $ B $ be two finite-dimensional algebras over an algebraically closed field. Suppose that $ A $ and $ B $ are stably equivalent of Morita type; we prove that $ A $ satisfies the Auslander–Reiten conjecture (resp. Gorenstein projective conjecture, strong Nakayama conjecture, Auslander–Gorenstein conjecture, Nakayama conjecture, Gorenstein symmetric conjecture) if and only if $ B $ does so. This can provide new classes of algebras satisfying homological conjectures, and we give an example to illustrate it.
Citation: Juxiang Sun, Guoqiang Zhao. Homological conjectures and stable equivalences of Morita type[J]. AIMS Mathematics, 2025, 10(2): 2589-2601. doi: 10.3934/math.2025120

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Abstract

Let $ A $ and $ B $ be two finite-dimensional algebras over an algebraically closed field. Suppose that $ A $ and $ B $ are stably equivalent of Morita type; we prove that $ A $ satisfies the Auslander–Reiten conjecture (resp. Gorenstein projective conjecture, strong Nakayama conjecture, Auslander–Gorenstein conjecture, Nakayama conjecture, Gorenstein symmetric conjecture) if and only if $ B $ does so. This can provide new classes of algebras satisfying homological conjectures, and we give an example to illustrate it.

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