Yoshizawa investigated when local cohomology modules have an annihilator that does not depend on the choice of defining ideal. In this paper we refine his results and investigate relationship between annihilators restricted flat dimensions.

We define a new category analogous to ${\bf FI}$ for the $0$-Hecke algebra $H_n(0)$ called category, $\mathcal{H}$, indexing sequences of representations as $n$ varies under suitable compatibility conditions. establish type representation stability in this setting and prove it is implied by being finitely generated $\mathcal{H}$-module. then provide examples $\mathcal{H}$-modules discuss furthe...