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

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

2003-12-09

arxiv.org/abs/math/0312190v6

Advances in Mathematics 203 (2006), 194-255.

Mathematics

Algebraic Geometry

56 pages, LaTeX. (v6) Minor changes, now in final form

Scientific paper

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.

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