Scattering diagrams for Artin algebras

Abstract

For an arbitrary Artin algebra $A$, we construct a minimal and consistent scattering diagram by approximating its module category $\mathrm{mod}\,A$ using the subcategories $(\mathrm{mod}\,A)_\ell$ of modules of length at most $\ell \in \mathbb{N}$. We prove that each subcategory $(\operatorname{mod}A)_\ell$ possesses a well-behaved lattice of torsion classes, a finite wall-and-chamber structure $\mathfrak{D}_\ell(A)$ and an associated picture group $G_\ell(A)$ with a natural categorical interpretation. Using these properties, we build a finite, minimal, consistent scattering diagram for each $\ell \in \mathbb{N}$. Passing to the inverse limit, we establish the existence of a minimal consistent scattering diagram for $A$. In particular, when $A$ is a finite-dimensional algebra over $\mathbb{C}$, our inverse limit construction is canonically isomorphic to Bridgeland's stability scattering diagram.