Configurations in abelian categories. I: Basic properties and moduli stacks
Let \((I, \leq)\) be a finite partially ordered set and let \(J, K\) denote subsets of \(I\). The author defines an \((I, \leq)\)-configuration in an abelian category \(A\) as a finite collection of objects \(\sigma(J)\) and morphisms \(\sigma(J) \to \sigma(K)\) in \(A\) satisfying certain axioms. In this paper, basic theory of configurations is developed in detail and a Jordan-Hölder type theorem is proved. For a \(k\)-linear abelian category \(A\), \(k\) being an algebraically closed field, moduli stacks of objects of \(A\) and of configurations in \(A\) and certain substacks of them with fixed classes in \(K(A)\) are constructed under a couple of assumptions. These are locally finite algebraic (Artin) \(k\)-stacks. Some natural morphisms among these stacks are shown to be representable or of finite type. Finally these stacks are studied for some interesting examples of \(A\), viz. the category of coherent sheaves on a projective \(k\)-scheme, category of representations of a quiver \(Q\) and the category of representations of finite dimensional \(k\)-algebras. The paper is the first in a long series of papers on configuration of abelian categories. The subsequet works deal with Ringel-Hall algebras, moduli of (semi)stable objects in configurations and variation of stability.
scientific article