Big Finitistic Dimensions of Radical-Cube-Zero Algebras

Abstract

We construct, over any field, a ten-dimensional algebra $A$ with $(\operatorname{rad}A)^3=0$ such that $\operatorname{findim}A=2$, $\operatorname{Findim}A=\infty$, and $\operatorname{Findim}(A^{\mathrm{op}})=0$. 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.