  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 $edell$, the sub-derived delooping level $subddell$, and the
derived delooping level $ddell$. They are all better upper bounds for the
opposite Findim. Precisely, we prove \[ Findim\Lambda^{op} = edell\Lambda \leq
ddell\Lambda \text{ (or $subddell\Lambda$)} \leq dell\Lambda \] and provide
examples where the last inequality is strict (including the recent example from
[Kershaw and Rickard] where $dell\Lambda=\infty$, but
$ddell\Lambda=1=Findim\Lambda^{op}$).
  We further enhance the connection between the findim conjecture to tilting
theory by showing finitely generated modules with finite derived delooping
level form a torsion-free class $F$. Therefore, studying the corresponding
torsion pair $(T, F)$ 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.
