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 $(\mathrm{mod}A{)}_{\ell }$ 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.

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