Configurations in abelian categories - I: Basic properties and moduli stacks

This is the first in a series of papers on configurations in an abelian category A. Given a finite partially ordered set (I, less than or similar to), an (I, less than or similar to)-configuration (sigma, iota, pi) is a finite collection of objects sigma(J) and morphisms iota(J, K) or pi(J, K) : sigma(J) -> sigma(K) in A satisfying some axioms, where J, K are subsets of I. Configurations describe how an object X in A decomposes into subobjects, and are useful for studying stability conditions on A. We define and motivate the idea of configurations, and explain some natural operations upon them-subconfigurations, quotient configurations, substitution, refinements and improvements. Then we study moduli spaces of (I, less than or similar to)-configurations in A, and natural morphisms between them, using the theory of Arlin stacks. We prove well-behaved moduli stacks exist when A is the abelian category of coherent sheaves on a projective scheme P, or of representations of a quiver Q. In the sequels, given a stability condition (tau, T, <=) on A, we will show the moduli spaces of tau-(semi)stable objects or configurations are constructible subsets in the moduli stacks of all objects or configurations. We associate infinite-dimensional algebras of constructible functions to a quiver Q using the method of Ringel-Hall algebras, and define systems of invariants of P that 'count' tau-(semi)stable coherent sheaves on P and satisfy interesting identities. (C) 2005 Elsevier Inc. All rights reserved.