2 problems

Let $c$ be a positive integer. Let $\mathbf{M}$ be a finitely generated $\operatorname{OI}$-module over $\mathbf{P}=({\mathbf{X}}^{\operatorname{OI},1})^{\otimes c}$, or a finitely…

Let $V$ be an ${\mathbf{OI}}$-module, and let $r$ be a nonnegative integer. Say that $V$ has level at most $r$ as in the cited source, and let $t_i(V)$ denote the relevant homologi…