Derived delooping levels and finitistic dimension

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 $$\mathrm{Findim}\,\Lambda^{\mathrm{op}} = \mathrm{edell}\,\Lambda \leq \mathrm{ddell}\,\Lambda \text{ (or $\mathrm{subddell}\,\Lambda$)} \leq \mathrm{dell}\,\Lambda$$ and provide examples where the last inequality is strict (including the recent example from [16] where $\mathrm{dell}\,\Lambda=\infty$, but $\mathrm{ddell}\, \Lambda = 1 =\mathrm{Findim}\, \Lambda^{\mathrm{op}}$). 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 $\phi$-dimension $\phi\dim$, a popular upper bound for findim, and give another sufficient condition for the findim conjecture.