Homology of FI-modules
type: Item
instance of: scholarly article
title: Homology of FI-modules
zbMATH Open document ID: 1371.18012
author: Thomas Church
author: Jordan S. Ellenberg
published in: Geometry \& Topology
publication date: 2017-06-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1506.01022
review text: 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.
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 13C60
zbMATH DE Number: 6726524
zbMATH Keywords: FI-modules
zbMATH Keywords: homology
zbMATH Keywords: Castelnuovo-Mumford regularity
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
OpenAlex ID: W2231551317
arXiv ID: 1506.01022
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DOI: 10.2140/GT.2017.21.2373
description: scientific article