  We study the derived Hecke action at $p$ on the ordinary $p$-adic cohomology
of arithmetic subgroups of semisimple groups $\mathrm G(\mathbb Q)$, i.e., we
study the derived version of Hida's theory for ordinary Hecke algebras. This is
the analog at $\ell=p$ of derived Hecke actions studied by Venkatesh in the
tame case. We show that properties of the derived Hecke action at $p$ are
related to deep conjectures in Galois cohomology which are higher analogs of
the classical Leopoldt conjecture.
