Derived delooping levels and finitistic dimension
Ruoyu Guo, Kiyoshi Igusa
TL;DR
The paper tackles the finitistic dimension conjecture by introducing three new invariants—effective delooping level (edell), derived delooping level (ddell), and sub-derived delooping level (subddell)—as tighter upper bounds for Findim$^{op}$. It proves the foundational inequality $\mathrm{Findim}\,\Lambda^{\mathrm{op}} = \mathrm{edell}\,\Lambda \leq \mathrm{ddell}\,\Lambda \leq \mathrm{dell}\,\Lambda$ (with $\mathrm{subddell}$ as a possible bound) and shows these invariants can strictly improve upon the original delooping level in some cases, including a monomial algebra example where $\mathrm{dell}$ is infinite but $\mathrm{ddell}=1$. The work connects these invariants to tilting theory by showing modules of finite ddell form a torsion-free class, leading to a torsion pair that informs the little finitistic dimension, and it relates the dell-type invariants to the φ-dimension, providing a practical finiteness criterion via a finite set $T_{\Lambda}$. The paper also analyzes the derived invariants in concrete examples (notably a Kershaw–Rickard-type algebra) showing finite ddell even when dell is infinite, and it proposes a sufficient condition for Findim finiteness that recovers known results for monomial algebras while offering new perspectives through derived-category and tilting-theoretic viewpoints. Overall, these invariants offer a refined toolkit for approaching Findim and illuminate connections between homological dimensions, tilting, and derived categories.
Abstract
In this paper, we develop new ideas regarding the finitistic dimension conjecture, or the findim conjecture for short. Specifically, we improve upon the delooping level by introducing three new invariants called the effective delooping level $\mathrm{edell}$, the sub-derived delooping level $\mathrm{subddell}$, and the derived delooping level $\mathrm{ddell}$. They are all better upper bounds for the opposite Findim. Precisely, we prove \[ \mathrm{Findim}\,Λ^{\mathrm{op}} = \mathrm{edell}\,Λ\leq \mathrm{ddell}\,Λ\text{ (or $\mathrm{subddell}\,Λ$)} \leq \mathrm{dell}\,Λ\] and provide examples where the last inequality is strict (including the recent example from [16] where $\mathrm{dell}\,Λ=\infty$, but $\mathrm{ddell}\, Λ= 1 =\mathrm{Findim}\, Λ^{\mathrm{op}}$). We further enhance the connection between the findim conjecture and tilting theory by showing finitely generated modules with finite derived delooping level form a torsion-free class $\mathcal{F}$. Therefore, studying the corresponding torsion pair $(\mathcal{T}, \mathcal{F})$ will shed more light on the little finitistic dimension. Lastly, we relate the delooping level to the $φ$-dimension $φ\dim$, a popular upper bound for findim, and give another sufficient condition for the findim conjecture.