Geometric models for Ginzburg algebras of n-angulated surfaces via local-to-global principles

Merlin Christ

Published 2021-07-21Version 1

We relate the derived category of a relative Ginzburg algebra of an n-angulated surface to the geometry of the underlying surface. Results include the description of a subset of the objects in the derived category in terms of curves in the surface and their Hom's in terms of intersection. By using the description of such a derived category as the global sections of perverse schober, we arrive at the geometric model through gluing local data. Nearly all results also hold for the perverse schober defined over any commutative ring spectrum. As a direct application of the geometric model, we categorify the extended mutation matrices of a class of cluster algebras with coefficients, associated to multi-laminated marked surfaces by Fomin-Thurston.