Configurations in abelian categories. I: Basic properties and moduli stacks
type: Item
instance of: scholarly article
title: Configurations in abelian categories. I: Basic properties and moduli stacks
zbMATH Open document ID: 1102.14009
published in: Advances in Mathematics
publication date: 2006-08-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0312190
review text: Let \((I, \leq)\) be a finite partially ordered set and let \(J, K\) denote subsets of \(I\). The author defines an \((I, \leq)\)-configuration in an abelian category \(A\) as a finite collection of objects \(\sigma(J)\) and morphisms \(\sigma(J) \to \sigma(K)\) in \(A\) satisfying certain axioms. In this paper, basic theory of configurations is developed in detail and a Jordan-Hölder type theorem is proved. For a \(k\)-linear abelian category \(A\), \(k\) being an algebraically closed field, moduli stacks of objects of \(A\) and of configurations in \(A\) and certain substacks of them with fixed classes in \(K(A)\) are constructed under a couple of assumptions. These are locally finite algebraic (Artin) \(k\)-stacks. Some natural morphisms among these stacks are shown to be representable or of finite type. Finally these stacks are studied for some interesting examples of \(A\), viz. the category of coherent sheaves on a projective \(k\)-scheme, category of representations of a quiver \(Q\) and the category of representations of finite dimensional \(k\)-algebras. The paper is the first in a long series of papers on configuration of abelian categories. The subsequet works deal with Ringel-Hall algebras, moduli of (semi)stable objects in configurations and variation of stability.
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 5043535
zbMATH Keywords: moduli stack
zbMATH Keywords: coherent sheaves
zbMATH Keywords: quivers
Wikidata QID: Q56990991
author: Dominic Joyce
reviewed by: Usha N. Bhosle
MaRDI profile type: Publication
OpenAlex ID: W2092350576
arXiv ID: math/0312190
cites work: Q3999421
cites work: Q4783713
cites work: Algebraic stacks
cites work: Q5645518
cites work: Q3313974
cites work: FRAMED MODULES AND THEIR MODULI
cites work: MODULI OF REPRESENTATIONS OF FINITE DIMENSIONAL ALGEBRAS
cites work: Q4935967
cites work: Q3215398
cites work: Space of unitary vector bundles on a compact Riemann surface
DOI: 10.1016/J.AIM.2005.04.008
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description: scientific article