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

Abstract

This is the first in a series of papers on configurations in an abelian category A . Given a finite partially ordered set ( I , ≼ ) , an ( I , ≼ ) - 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, 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 , ≼ ) -configurations in A , and natural morphisms between them, using the theory of Artin 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 ( τ , T , ⩽ ) on A , we will show the moduli spaces of τ -(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’ τ -(semi)stable coherent sheaves on P and satisfy interesting identities.