Filtrations and homological degrees of FI-modules
type: Item
instance of: scholarly article
title: Filtrations and homological degrees of FI-modules
zbMATH Open document ID: 1371.13015
author: Liping Li
author: Nina Yu
published in: Journal of Algebra
publication date: 2016-12-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1511.02977
review text: Let \(\mathcal{FI}\) be the category introduced by \textit{T. Church} and \textit{B. Farb} [Adv. Math. 245, 250--314 (2013; Zbl 1300.20051)] whose objects are finite sets and morphisms injections between, and write \(\mathcal{C}\) for the skeletal category. A {\(\mathcal{C}\)-module} is a covariant functor \(V\) from \(\mathcal{C}\) to \(\Bbbk\)-Mod, the category of left \(\Bbbk\)-modules for a Noetherian ring \(\Bbbk\). The authors use the shift functor \(\Sigma : \mathcal{C}\text{-Mod}\to \mathcal{C}\text{-Mod}\) to investigate homological degrees and special filtrations of \(\mathcal{FI}\)-modules to obtain a homological characterization of \(\sharp\)-\textit{filtered modules} defined by \textit{R. Nagpal} [``FI-modules and the cohomology of modular representations of symmetric groups'', \url{arXiv:1505.04294}]. Then, this filtration is applied and a result of Church and Farb [loc. cit.] is used to get other upper bounds for projective dimensions and homological degrees of finitely generated \(\mathcal{FI}\)-modules.
reviewed by: Marek Golasiński
Mathematics Subject Classification ID: 13C60
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 6668109
zbMATH Keywords: \(FI\)-module
zbMATH Keywords: filtration
zbMATH Keywords: homological degree
zbMATH Keywords: projective dimension
MaRDI profile type: Publication
OpenAlex ID: W2963209312
arXiv ID: 1511.02977
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DOI: 10.1016/J.JALGEBRA.2016.11.019
description: scientific article