Filtrations and homological degrees of FI-modules

Let k be a commutative Noetherian ring. In this paper we consider ♯-filtered modules of the category FI firstly introduced in [12]. We show that a finitely generated FI-module V is ♯-filtered if and only if its higher homologies all vanish, and if and only if a certain homology vanishes. Using this homological characterization, we characterize finitely generated C-modules V whose projective dimension pd(V) is finite, and describe an upper bound for pd(V). Furthermore, we give a new proof for the fact that V induces a finite complex of ♯-filtered modules, and use it as well as a result of Church and Ellenberg in [1] to obtain another upper bound for homological degrees of V.