PreprintCC BY 4.0arXiv
Igusa--Todorov Algebras and the Auslander--Reiten Conjecture
Abstract
We prove that every Igusa--Todorov Artin algebra satisfies the Auslander--Reiten conjecture. More precisely, let V be an n-Igusa--Todorov witness, and let t be the number of isomorphism classes of nonprojective indecomposable summands of V. If a finitely generated module M satisfies ExtAi(M,A)=0 for every i>0 and ExtAq(M,M)=0 for 1≤ q≤ 2t+1, then M is projective. As applications, algebras of representation dimension at most three and algebras satisfying J2m+1=0 for which A/Jm has finite representation type satisfy the Auslander--Reiten conjecture.
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BibTeX
@misc{zhang2026igusatodorov,
title = {{Igusa--Todorov Algebras and the Auslander--Reiten Conjecture}},
author = {Xiaojin Zhang and Panyue Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14453},
eprint = {2609.14453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14453},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Igusa--Todorov Algebras and the Auslander--Reiten Conjecture
AU - Xiaojin Zhang
AU - Panyue Zhou
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.14453
UR - https://arxiv.org/abs/2609.14453
AB - We prove that every Igusa--Todorov Artin algebra satisfies the Auslander--Reiten conjecture. More precisely, let V be an n-Igusa--Todorov witness, and let t be the number of isomorphism classes of nonprojective indecomposable summands of V. If a finitely generated module M satisfies Ext_A^i(M,A)=0 for every i>0 and Ext_A^q(M,M)=0 for 1≤ q≤ 2t+1, then M is projective. As applications, algebras of representation dimension at most three and algebras satisfying J^2m+1=0 for which A/J^m has finite representation type satisfy the Auslander--Reiten conjecture.
ER -