Homology of FI-modules
FI denotes the category of finite sets and injections; an FI-module W is a functor \(W: FI\longrightarrow\mathbb Z\text{ Mod}\). Given a finite set \(T\), \(W_T\) stands for \(W(T)\). For every \(n\in\mathbb N\), \([n]\) stands for the set \(\{1,\dots,n\}\) and \(W_n\) stands for \(W_{[n]}=W([n])\). This long paper contains a number of result of independent interest. The key result seems to be Theorem E: Theorem. Let \(M\) be a torsion-free FI-module generated in degree at most \(k\), and let \(V\subseteq M\) be a sub-FI-module generated in degree at most \(d\). Then for all \(n > \min(k; d)+d\) and any \(a \leq n\), \(V_n \cap (M_{[n]-\{1\}}+\dots+M_{[n]-\{a\}}) = (V_{[n]-\{1\}}+\dots+V_{[n]-\{a\}})\). The theorem is valid for every \(d\geq 0\) and \(k\geq 0\), although the cases of primary interest are when \(k<d\). Theorem E may be stated without \(M\) and it is connected to homology. The authors show that \(M\) is a free \(FI\)-module. It has a natural homological interpretation as a bound on the degree of vanishing of a certain derived functor applied to \(M/V\). This is how one can connect this theorem to bounds on regularity, a subject of Theorem A. The authors prove an explicit and sharp upper bound for the Castelnuovo-Mumford regularity of an FI-module in terms of the degrees of its generators and relations. This is then used to refine a result of \textit{A. Putman} [Invent. Math. 202, No. 3, 987--1027 (2015; Zbl 1334.20045)] on the stability of homology of congruence subgroups, extending his theorem to previously excluded small characteristics and to integral homology while maintaining explicit bounds for the stable range.
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