Ginzburg algebras of triangulated surfaces and perverse schobers

Ginzburg algebras associated to triangulated surfaces provide a means to categorify the cluster algebras of these surfaces. As shown by Ivan Smith, the finite derived category of such a Ginzburg algebra can be embedded into the Fukaya category of the total space of a Lefschetz fibration over the sur...

Full description

Saved in:
Bibliographic Details
Published inForum of mathematics. Sigma Vol. 10
Main Author Christ, Merlin
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.01.2022
Subjects
Algebra
Combinatorics
Geometry
Gluing
Triangulation
18N60
18G80
18N25
Online AccessGet full text

Cover

More Information
Summary:Ginzburg algebras associated to triangulated surfaces provide a means to categorify the cluster algebras of these surfaces. As shown by Ivan Smith, the finite derived category of such a Ginzburg algebra can be embedded into the Fukaya category of the total space of a Lefschetz fibration over the surface. Inspired by this perspective, we provide a description of the unbounded derived category in terms of a perverse schober. The main novelty is a gluing formalism describing the Ginzburg algebra as a colimit of certain local Ginzburg algebras associated to discs. As a first application, we give a new construction of derived equivalences between these Ginzburg algebras associated to flips of an edge of the triangulation. Finally, we note that the perverse schober as well as the resulting gluing construction can also be defined over the sphere spectrum.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2022.1