Configurations in abelian categories. I. Basic properties and moduli stacks

Abstract

This is the first in a series of papers math.AG/0503029, math.AG/0410267, math.AG/0410268 on "configurations" in an abelian category A. Given a finite partially ordered set (I,<), an (I,<)-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) satisfying some axioms, where J,K are subsets of I. Configurations describe how an object X in A decomposes into subobjects, and are especially useful for studying stability conditions on A. This paper defines and motivates the idea of configurations, and explains some natural operations upon them -- subconfigurations, quotient configurations, refinements, improvements and substitution. Then we study moduli spaces of (I,<)-configurations in A, using the theory of Artin stacks. We prove well-behaved moduli stacks exist when A is an abelian category of coherent sheaves or vector bundles on a projective K-scheme P, or of representations of a quiver Q. We define many natural 1-morphisms between the moduli stacks, some of which are representable or of finite type. The sequels will apply these results to construct and study infinite-dimensional algebras associated to a quiver Q, and to define systems of invariants of a projective K-scheme P that "count" (semi)stable coherent sheaves and satisfy interesting identities.