Noetherian approximation of algebraic spaces and stacks
type: Item
instance of: scholarly article
title: Noetherian approximation of algebraic spaces and stacks
zbMATH Open document ID: 1308.14006
author: David Rydh
published in: Journal of Algebra
publication date: 2014-11-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/0904.0227
review text: This paper, as its name suggests, deals with the problem of approximating certain structures such as algebraic spaces, schemes and stacks. The fundamental example as to what approximation means here is the following: Every commutative ring \(R\) can be written as a direct limit of its subrings that are finitely generated \(\mathbb Z\)-algebras and every \(R\)-module \(M\) can be written as the direct limit of its finitely generated \(R\)-submodules. This means that every affine scheme \(X\) is an inverse limit of affine schemes of finite type over \(\mathrm{Spec}\mathbb Z\) and every quasi-coherent sheaf on \(X\) is direct limit of quasi-coherent sheaves of finite type. In the paper it is shown for example that every quasi- compact and quasi-separated Deligne-Mumford stack \(X\) is an inverse limit of Deligne-Mumford stacks of finite type over \(\mathrm{Spec}\mathbb Z\). Approximation here is in contrast with standard limit results which are relative: given an inverse limit \(X = \underset\longleftarrow\lim_{\lambda} X_{\lambda}\) describe finitely presented objects over \(X\) in terms of finitely presented objects over \(X_{\lambda}\) for sufficiently large \(\lambda\). The following definitions are fundamental for understanding the paper: An algebraic stack \(X\) is said to have the completeness property if every quasicoherent sheaf on \(X\) is a filtered direct limit of finitely presented sheaves or, equivalently, if the abelian category \(QCoh(X)\) is compactly generated. Such an stack is called \textit{pseudo-noetherian} if it is quasi-compact, quasi-separated and \(X'\) has the completeness property for every finitely presented morphism \(X' \to X\) of algebraic stacks. For \(S\) a pseudo-noetherian stack and \(X \to S\) a morphism of stacks, an approximation of \(X\) over \(S\) is a finitely presented \(S\)-stack \(X_0\) together with an affine \(S\)-morphism \(X \to X_0\). \(X/S\) can be approximated if there is such an approximation over \(S\). A morphism \…
reviewed by: Mohajer Abolfazl
Mathematics Subject Classification ID: 14A20
zbMATH DE Number: 6370246
zbMATH Keywords: noetherian approximation
zbMATH Keywords: algebraic spaces
zbMATH Keywords: algebraic stacks
zbMATH Keywords: Chevalley's theorem
zbMATH Keywords: Serre's theorem
zbMATH Keywords: global quotient stacks
zbMATH Keywords: global type
zbMATH Keywords: basic stacks
MaRDI profile type: Publication
OpenAlex ID: W1977252077
arXiv ID: 0904.0227
cites work: Tame stacks in positive characteristic
cites work: Good moduli spaces for Artin stacks
cites work: Q5459656
cites work: Nagata compactification for algebraic spaces
cites work: Brauer groups and quotient stacks
cites work: Éléments de géométrie algébrique. I: Le langage des schémas. II: Étude globale élémentaire de quelques classe de morphismes. III: Étude cohomologique des faisceaux cohérents (première partie)
cites work: Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie). Rédigé avec la colloboration de J. Dieudonné
cites work: Q5602063
cites work: Algebraic spaces
cites work: Flattening stratification and the stack of partial stabilizations of prestable curves
cites work: Q4935967
cites work: On compactification of schemes
cites work: On proper coverings of Artin stacks
cites work: \(\underline{\Hom}\)-stacks and restriction of scalars
cites work: Toric vector bundles, branched covers of fans, and the resolution property
cites work: Approximation des schémas en groupes, quasi compacts sur un corps
cites work: Q5550339
cites work: Critères de platitude et de projectivité. Techniques de ''platification'' d'un module. (Criterial of flatness and projectivity. Technics of ''flatification of a module.)
cites work: Submersions and effective descent of étale morphisms
cites work: The canonical embedding of an unramified morphism in an étale morphism
cites work: Étale dévissage, descent and pushouts of stacks
cites work: Representability of Hilbert Schemes and Hilbert Stacks of Points
cites work: Stacks Project Expository Collection
cites work: Equivariant completion. II
cites work: Relative Riemann-Zariski spaces
cites work: Q3356420
cites work: The resolution property for schemes and stacks
DOI: 10.1016/J.JALGEBRA.2014.09.012
description: scientific article