Filtrations and homological degrees of FI-modules (Q729742)

Let \(\mathcal{FI}\) be the category introduced by \textit{T. Church} and \textit{B. Farb} [Adv. Math. 245, 250--314 (2013; Zbl 1300.20051)] whose objects are finite sets and morphisms injections between, and write \(\mathcal{C}\) for the skeletal category. A {\(\mathcal{C}\)-module} is a covariant functor \(V\) from \(\mathcal{C}\) to \(\Bbbk\)-Mod, the category of left \(\Bbbk\)-modules for a Noetherian ring \(\Bbbk\). The authors use the shift functor \(\Sigma : \mathcal{C}\text{-Mod}\to \mathcal{C}\text{-Mod}\) to investigate homological degrees and special filtrations of \(\mathcal{FI}\)-modules to obtain a homological characterization of \(\sharp\)-\textit{filtered modules} defined by \textit{R. Nagpal} [``FI-modules and the cohomology of modular representations of symmetric groups'', \url{arXiv:1505.04294}]. Then, this filtration is applied and a result of Church and Farb [loc. cit.] is used to get other upper bounds for projective dimensions and homological degrees of finitely generated \(\mathcal{FI}\)-modules.