The notion of FI-modules was introduced by T. Church et al. [Duke Math. J. 164, No. 9, 1833–1910 (2015; Zbl 1339.55004)]. Let FI be the category of finite sets and injections. It is equivalent to the category such that

\(\bullet\)

objects: \([n] = \{1, 2, \dots, n\}\) \((n=0,1,\dots)\),

\(\bullet\)

\(\mathrm{Hom}([m], [n]) = \{f : [m] \to [n] \mid f\text{ is injective}\}\).

An FI-module over a Noetherian ring \(k\) is a functor from FI to the category of \(k\)-modules. That is, an FI-module \(M\) consists of a family of \(k\)-modules \(\{M_n \mid n = 0, 1, \dots\}\) and \(\binom{n}{m}\) \(k\)-homomorphisms \([m] \to [n]\) for any \(m \leq n\). We can identify \(M\) a graded \(k\)-module \(\bigoplus M_n\).
On the other hand, the Castelnuovo-Munford regularity is an important invariant of finitely generated graded module \(M\) over a polynomial ring \(S\). It is defined by local cohomology functor but it also defined by torsion functor. Indeed, the regularity of \(M\) equals to \[ \max\{\max\{n \mid H_{\mathfrak m}^i(M)_n \ne 0\}+i \mid i \geq 0\} = \max\{\min\{n \mid \text{Tor}_i(S/\mathfrak m, M)_n \ne 0\}-i \mid i \geq 0\}. \tag{\#} \] In the paper under review, the authors show an analogue. They define local cohomology functor and torsion functor of FI-modules and proved the similar equality to (#).