Big Finitistic Dimensions of Radical-Cube-Zero Algebras
Abstract
We construct, over any field, a ten-dimensional algebra A with (radA)3=0 such that findimA=2, FindimA=∞, and Findim(Aop)=0. An explicit inverse-syzygy construction gives countably generated right modules of every positive finite projective dimension, disproving the big finitistic dimension conjecture. At each step we identify the full projective-cover kernel, and nonzero Ext detects the exact resolution length. For this algebra, we classify the modules of projective dimension at most one that embed in radicals of projective modules, with no generation restriction, in terms of invertible linear operators. We determine all dimension vectors of finite-dimensional modules of projective dimension two and classify those attaining the sharp dimension bounds, including all fourteen-dimensional modules of projective dimension two. For arbitrary Artin algebras, we give recursive sufficient criteria that control finite projective resolutions through radical-annihilated ideals and suitable corners. For finite-dimensional elementary radical-cube-zero algebras, we prove finiteness of the big finitistic dimension in the two-vertex case with one loop at each vertex and obtain multiplication-map criteria for bounds of one or two. Finally, an exact representation embedding yields wildness of the projective-dimension-two class and, over every field, a superdecomposable pure-injective module in that class.
Related
- Thick subcategories over weakly symmetric algebras with radical cube zero
- The Auslander-Reiten conjecture for algebras with radical cube zero
- Syzygy-Quiver Formulas and Finitistic Dimensions of Special Biserial Algebras
- Local-Cohen-Macaulay DG-rings with unequal finitistic projective dimensions
- Derived Delooping Levels of One-Point Extensions and Finitistic Dimensions
BibTeX
@misc{chen2026finitistic,
title = {{Big Finitistic Dimensions of Radical-Cube-Zero Algebras}},
author = {Liang Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00433},
eprint = {2610.00433},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00433},
}
RIS
TY - UNPB
TI - Big Finitistic Dimensions of Radical-Cube-Zero Algebras
AU - Liang Chen
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00433
UR - https://arxiv.org/abs/2610.00433
AB - We construct, over any field, a ten-dimensional algebra A with (radA)^3=0 such that findimA=2, FindimA=∞, and Findim(A^op)=0. An explicit inverse-syzygy construction gives countably generated right modules of every positive finite projective dimension, disproving the big finitistic dimension conjecture. At each step we identify the full projective-cover kernel, and nonzero Ext detects the exact resolution length. For this algebra, we classify the modules of projective dimension at most one that embed in radicals of projective modules, with no generation restriction, in terms of invertible linear operators. We determine all dimension vectors of finite-dimensional modules of projective dimension two and classify those attaining the sharp dimension bounds, including all fourteen-dimensional modules of projective dimension two. For arbitrary Artin algebras, we give recursive sufficient criteria that control finite projective resolutions through radical-annihilated ideals and suitable corners. For finite-dimensional elementary radical-cube-zero algebras, we prove finiteness of the big finitistic dimension in the two-vertex case with one loop at each vertex and obtain multiplication-map criteria for bounds of one or two. Finally, an exact representation embedding yields wildness of the projective-dimension-two class and, over every field, a superdecomposable pure-injective module in that class.
ER -