Configurations in abelian categories. III: Stability conditions and identities
type: Item
instance of: scholarly article
title: Configurations in abelian categories. III: Stability conditions and identities
zbMATH Open document ID: 1134.14007
published in: Advances in Mathematics
publication date: 2007-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0410267
review text: The author has a series of four long papers on configurations in an abelian category \(A\), this one being the third in the series [for the other parts, cf. Adv. Math. 203, 194--255 (2006; Zbl 1102.14009), Adv. Math. 210, 635--706 (2007; Zbl 1119.14005); Adv. Math. 217, No. 1, 125--204 (2008; Zbl 1134.14008)]. For a finite partially ordered set \((I,\preceq)\) in \(A\), an \((I,\preceq)\)-configuration is a finite collection of objects in \(I\) and morphisms between them satisfying certain conditions. Let \(K(A)\) be the Grothendieck group of \(A\) and \(C(A)\subset K(A)\) the collection of classes in \(K(A)\) of all non-zero objects in \(A\). Denote by \(Obj_A\) and \(Obj_A^{\alpha}\) respectively the moduli stacks of objects in \(A\) and of objects in \(A\) with class \(\alpha\) in \(K(A)\). Generalizing \textit{A. Rudakov}'s stability conditions [J. Algebra, 197, 231--245 (1997; Zbl 0893.18007)], the author defines a (weak) stability condition to be a triple \((\tau, T, \leq)\) where \((T,\leq)\) is a totally ordered set and \(\tau: C(A) \to T\) is a map with certain properties. Notions of \(\tau\)-(semi)stability, \(\tau\)-unstability for objects in \(A\) are introduced. For example, slope functions for quiver representations and Gieseker stability of coherent sheaves are stability conditions while \(\mu\)-stability and purity of coherent sheaves are examples of weak stability conditions. It is shown that if \((\tau, T, \leq)\) is a (permissible) stability condition, then each \(\tau\)-semistable object \(X\) has a (non-unique) filtration with unique \(\tau\)-stable factors (successive quotients) \(S_k\) satisfying \(\tau(S_k)= \tau(X)\). If \((\tau, T, \leq)\) is a weak stability condition, then each \(X\) has a unique Harder-Narasimhan filtration with \(\tau\)-semistable factors \(S_k\) such that \(\tau(S_1)>\cdots >\tau(S_n=X)\). The author forms moduli spaces \(Obj_{ss}^{\alpha}, Obj_{st}^{\alpha},Obj_{si}^{\alpha}\), of \(\tau\)-semistable, \(\tau\)-stable …
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 18F20
zbMATH DE Number: 5196010
zbMATH Keywords: configuration
zbMATH Keywords: abelian category
zbMATH Keywords: stability condition
zbMATH Keywords: moduli spaces
author: Dominic Joyce
reviewed by: Usha N. Bhosle
MaRDI profile type: Publication
OpenAlex ID: W2021993770
Wikidata QID: Q56990978
arXiv ID: math/0410267
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DOI: 10.1016/J.AIM.2007.04.002
description: scientific article