  Suppose $F$ is a finite extension of $\mathbb{Q}_p$, $G$ is the group of
$F$-points of a connected reductive $F$-group, and $I_1$ is a pro-$p$-Iwahori
subgroup of $G$. We construct two spectral sequences relating derived functors
on mod-$p$ representations of $G$ to the analogous functors on Hecke modules
coming from pro-$p$-Iwahori cohomology. More specifically: (1) using results of
Ollivier--Vign\'eras, we provide a link between the right adjoint of parabolic
induction on pro-$p$-Iwahori cohomology and Emerton's functors of derived
ordinary parts; and (2) we establish a "Poincar\'e duality spectral sequence"
relating duality on pro-$p$-Iwahori cohomology to Kohlhaase's functors of
higher smooth duals. As applications, we calculate various examples of the
Hecke modules $\textrm{H}^i(I_1,\pi)$.
