Asymptotic behavior of representations of graded categories with inductive functors

In this paper we describe inductive machinery to investigate asymptotic behavior of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring k, via introducing inductive functors which generalize important properties of shift functors of FI-modules. In particular, a sufficient criterion for finiteness of Castelnuovo–Mumford regularity of finitely generated representations of these categories is obtained. As applications, we show that a few important infinite combinatorial categories appearing in representation stability theory (for example ${\mathrm{FI}}^d$, ${\mathrm{OI}}^d$, ${\mathrm{FI}}_d$, ${\mathrm{OI}}_d$) are equipped with inductive functors, and hence the finiteness of Castelnuovo–Mumford regularity of their finitely generated representations is guaranteed. We also prove that truncated representations of these categories have linear minimal resolutions by relative projective modules, which are precisely linear minimal projective resolutions when k is a field of characteristic 0.