Scattering diagrams for Artin algebras

2026/08/04 by Hipolito Treffinger
Mathematics · #math.RT #math.RA

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arxiv created 2026/08/04 · arxiv updated 2026/08/06

Abstract

For an arbitrary Artin algebra A, we construct a minimal and consistent scattering diagram by approximating its module category mod A using the subcategories (mod A)_ℓ of modules of length at most ℓ ∈ ℕ. We prove that each subcategory (modA)_ℓ possesses a well-behaved lattice of torsion classes, a finite wall-and-chamber structure \mathfrakD_ℓ(A) and an associated picture group G_ℓ(A) with a natural categorical interpretation. Using these properties, we build a finite, minimal, consistent scattering diagram for each ℓ ∈ 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 ℂ, our inverse limit construction is canonically isomorphic to Bridgeland's stability scattering diagram.

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