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Recollements for graded gentle algebras from spherical band objects

Pierre Bodin

TL;DR

The paper develops an explicit algebraic model for localizing the derived category of a graded gentle algebra along a spherical band object, corresponding to contracting a simple closed curve on the associated graded marked surface. It introduces graded pinched gentle algebras, proves a recollement describing the localization, and shows that in characteristic not equal to 2 this localized category is equivalent to the derived category of a pinched gentle algebra $\Lambda_{(\alpha,\beta)}$. A key technical advance is the use of Drinfeld quotients and formality results to compute the relevant Hom spaces, culminating in a precise correspondence between graded pinched gentle algebras and graded marked surfaces with conical singularities. The work unifies algebraic localization, DG-quotients, and surface models to produce a flexible framework for iterated localizations via pinching and contraction on the surface side, with potential applications to the study of partially wrapped Fukaya categories and their localizations.

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.

Recollements for graded gentle algebras from spherical band objects

TL;DR

The paper develops an explicit algebraic model for localizing the derived category of a graded gentle algebra along a spherical band object, corresponding to contracting a simple closed curve on the associated graded marked surface. It introduces graded pinched gentle algebras, proves a recollement describing the localization, and shows that in characteristic not equal to 2 this localized category is equivalent to the derived category of a pinched gentle algebra . A key technical advance is the use of Drinfeld quotients and formality results to compute the relevant Hom spaces, culminating in a precise correspondence between graded pinched gentle algebras and graded marked surfaces with conical singularities. The work unifies algebraic localization, DG-quotients, and surface models to produce a flexible framework for iterated localizations via pinching and contraction on the surface side, with potential applications to the study of partially wrapped Fukaya categories and their localizations.

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.
Paper Structure (13 sections, 24 theorems, 25 equations, 2 figures)

This paper contains 13 sections, 24 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Context
  3. Definitions and main results
  4. Graded pinched gentle algebras
  5. Recollections on DG categories
  6. Spectral sequences and Drinfeld quotients
  7. Admissible dissections and graded pinched gentle algebras
  8. Adapted admissible dissections
  9. Graded marked surfaces with conical singularities
  10. Formality of the quotient
  11. A quasi-equivalence
  12. Computation of $\mathrm{H^*}(\mathcal{A}/\mathcal{B})_{\circ}$
  13. Proof of the main results

Key Result

Theorem 1.3

Let $\Lambda'$ be a graded gentle algebra associated to the graded marked surface with admissible dissection $(S,M,\Delta,G)$. Let $\gamma$ be a simple closed curve on $S$ with winding number zero that does not enclose a subsurface containing only punctures, and let $B_{\gamma}$ be an associated ban Moreover, if the characteristic of $K$ is different from 2, there is:

Figures (2)

  • Figure 1.1: A marked surface with a simple closed curve (blue), and its marked surface with conical singularities obtained by contraction.
  • Figure 4.1: A marked surface with conical singularities and graded simple admissible dissection, and the splitting of its conical singularity.

Theorems & Definitions (44)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.3
  • Remark 1.4
  • Example 1.5
  • Definition 1.6
  • Remark 1.7
  • Example 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 34 more