PreprintCC BY 4.0arXiv
The Auslander-Reiten conjecture for algebras with radical cube zero
Abstract
Let A be a split finite-dimensional algebra over a field whose radical J satisfies J3=0, and let s be the number of isomorphism classes of simple A-modules. We prove that a non-projective module M with ExtAi(M,A)=0 for all i>0 has a non-zero self-extension in some degree between $1$ and 3s+1. In particular, A satisfies the Auslander--Reiten conjecture, which asserts that every self-orthogonal generator is projective. As a consequence, every finite-dimensional algebra over an algebraically closed field with radical cube zero satisfies the Auslander-Reiten conjecture.
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BibTeX
@misc{zhang2026auslanderreiten,
title = {{The Auslander-Reiten conjecture for algebras with radical cube zero}},
author = {Xiaojin Zhang and Panyue Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08679},
eprint = {2609.08679},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08679},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Auslander-Reiten conjecture for algebras with radical cube zero
AU - Xiaojin Zhang
AU - Panyue Zhou
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08679
UR - https://arxiv.org/abs/2609.08679
AB - Let A be a split finite-dimensional algebra over a field whose radical J satisfies J^3=0, and let s be the number of isomorphism classes of simple A-modules. We prove that a non-projective module M with Ext_A^i(M,A)=0 for all i>0 has a non-zero self-extension in some degree between $1$ and 3s+1. In particular, A satisfies the Auslander--Reiten conjecture, which asserts that every self-orthogonal generator is projective. As a consequence, every finite-dimensional algebra over an algebraically closed field with radical cube zero satisfies the Auslander-Reiten conjecture.
ER -