Configurations in abelian categories. III. Stability conditions and identities

Abstract

This is the third in a series on configurations in an abelian category A. Given a finite poset (I, {precedes or equal to}), an (I, {precedes or equal to})-configuration(σ, ι, π) is a finite collection of objects σ (J) and morphisms ι (J, K) or π (J, K) : σ (J) → σ (K) in A satisfying some axioms, where J, K are subsets of I. Configurations describe how an object X in A decomposes into subobjects. The first paper defined configurations and studied moduli spaces of configurations in A, using the theory of Artin stacks. It showed well-behaved moduli stacks ObjA, M (I, {precedes or equal to})A of objects and configurations in A exist when A is the abelian category coh (P) of coherent sheaves on a projective scheme P, or mod- K Q of representations of a quiver Q. The second studied algebras of constructible functions and stack functions on ObjA. This paper introduces (weak) stability conditions(τ, T, ≤) on A. We show the moduli spaces Objssα, Objsiα, Objstα (τ) of τ-semistable, indecomposable τ-semistable and τ-stable objects in class α are constructible sets in ObjA, and some associated configuration moduli spaces Mss, Msi, Mst, Mssb, Msib, Mstb (I, {precedes or equal to}, κ, τ)A constructible in M (I, {precedes or equal to})A, so their characteristic functions δssα, δsiα, δstα (τ) and δss, ..., δstb (I, {precedes or equal to}, κ, τ) are constructible. We prove many identities relating these constructible functions, and their stack function analogues, under pushforwards. We introduce interesting algebras Hτpa, Hτto, over(H, -)τpa, over(H, -)τto of constructible and stack functions, and study their structure. In the fourth paper we show Hτpa, ..., over(H, -)τto are independent of (τ, T, ≤), and construct invariants of A, (τ, T, ≤). © 2007 Elsevier Inc. All rights reserved.