Annihilation of cohomology, generation of modules and finiteness of derived dimension
type: Item
instance of: scholarly article
zbMATH Open document ID: 1368.13014
author: Ehsan Hakimian
author: Shokrollah Salarian
author: Ryo Takahashi
author: Abdolnaser Bahlekeh
published in: The Quarterly Journal of Mathematics
publication date: 2016-11-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1504.06163
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13C60
zbMATH DE Number: 6654443
zbMATH Keywords: cohomology annihilator
zbMATH Keywords: syzygy
zbMATH Keywords: singular locus
MaRDI profile type: Publication
OpenAlex ID: W2964246150
arXiv ID: 1504.06163
DOI: 10.1093/QMATH/HAW015
title: Annihilation of cohomology, generation of modules and finiteness of derived dimension
review text: Let \((R,\mathfrak{m},k)\) be a commutative noetherian local ring and \(mod~R\) the category of all finitely generated \(R\)-modules. Let \(\mathrm{ca}(R)\) be the cohomology annihilator of \(R\). The authors study properties of \(\mathrm{ca}(R)\) and the connections to singularity theory. They prove that there is some integer \(n\geq0\) such that \(\mathfrak{m}^n\subseteq \mathrm{ca}\;(R)\) if and only if the \(n\)th syzygies in \(mod~R\) are constructed from syzygies of the residue field o \(R\) in a finite number of extensions, direct sums and direct summands. The authors also show that, if \(d\) is the Krull dimension of \(R\) and \(M\) is an \(R\)-module that is locally free on the punctured spectrum of \(R\), then \(M\) is build out of syzygies of finite length by taking \(d\) extensions in \(mod~R\) up to finite direct sums and direct summands.
reviewed by: Anda-Georgina Olteanu
description: scientific article; zbMATH DE number 6654443