Recollements for graded gentle algebras from spherical band objects

TL;DR

This paper explores the localization of derived categories of graded gentle algebras via spherical band objects, leading to new algebra classes and geometric correspondences with graded marked surfaces.

Contribution

It introduces graded pinched gentle algebras and describes their relation to graded marked surfaces with conical singularities, expanding the understanding of derived category localizations.

Findings

Localization described as a recollement involving new graded algebras

Introduction of graded pinched gentle algebras as a generalization

Bijection between graded pinched gentle algebras and graded marked surfaces with conical singularities

Abstract

In this paper we study the localization of a derived category of a graded gentle algebra by a subcategory generated by a spherical band object. This object corresponds to a simple closed curve under the equivalence between the perfect derived category of the graded gentle algebra and the partially wrapped Fukaya category of the associated graded marked surface, as established by Haiden, Katzarkov and Kontsevich. We describe this localization as a recollement that involves the derived category of a new graded algebra given by quiver and relations. This leads us to the introduction of the class of graded pinched gentle algebras, a generalization of graded gentle algebras. We then show that these algebras are in bijection with graded marked surfaces with conical singularities. Moreover, under this correspondence the localization process amounts to the contraction of the closed curve.