PreprintCC BY 4.0arXiv
Self-Orthogonal τ-Tilting Modules and Tilting Modules III: Finitistic Dimension
Abstract
Let A be an Artin algebra and let T be a τ-tilting right A-module. Put C=A/AnnA(T). We prove that finite little finitistic dimension of Cop, together with a finite range of self-extension vanishing for T, forces T to be a classical $1$-tilting module. We also show that if the finitistic conjecture holds for all Artin algebras then the self-orthogonal τ-tilting conjecture holds for all Artin algebras. Moreover, a counterexample to the self-orthogonal τ-tilting conjecture will give a counterexample to the finitistic dimension conjecture. As an application, the self-orthogonal τ-tilting conjecture holds for Artin algebras with radical cube zero.
Related
- Self-Orthogonal τ-Tilting Modules and Tilting Modules II: Annihilator Separation
- Tilting bundles and modifying modules in relative dimension one
- Relative Derived Delooping Levels and τ-Tilting Modules
- Support τ-tilting modules over Morita context algebras: A bilateral approximation approach
- Transfer of Wakamatsu tilting modules along Frobenius extensions
BibTeX
@misc{zhang2026selforthogonal,
title = {{Self-Orthogonal $τ$-Tilting Modules and Tilting Modules III: Finitistic Dimension}},
author = {Xiaojin Zhang and Panyue Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00105},
eprint = {2610.00105},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00105},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Self-Orthogonal τ-Tilting Modules and Tilting Modules III: Finitistic Dimension
AU - Xiaojin Zhang
AU - Panyue Zhou
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00105
UR - https://arxiv.org/abs/2610.00105
AB - Let A be an Artin algebra and let T be a τ-tilting right A-module. Put C=A/Ann_A(T). We prove that finite little finitistic dimension of C^op, together with a finite range of self-extension vanishing for T, forces T to be a classical $1$-tilting module. We also show that if the finitistic conjecture holds for all Artin algebras then the self-orthogonal τ-tilting conjecture holds for all Artin algebras. Moreover, a counterexample to the self-orthogonal τ-tilting conjecture will give a counterexample to the finitistic dimension conjecture. As an application, the self-orthogonal τ-tilting conjecture holds for Artin algebras with radical cube zero.
ER -