Closure of graded gentle algebras under derived equivalence

Establish that the class of graded gentle algebras is closed under derived equivalence. Equivalently, show that for any graded smooth surface with stops S = (S, E, n), every formal generator of the partially wrapped Fukaya category W(S) is obtained from a formal dissection of S.

Background

For smooth surfaces, partially wrapped Fukaya categories admit descriptions via graded gentle algebras and formal generators coming from dissections. In the ungraded setting, closure under derived equivalence for gentle algebras is known (Schröer–Zimmermann), and recent work has provided derived invariants using line fields.

The authors posit the graded analogue: graded gentle algebras should remain within their class under derived equivalence, and geometrically, any formal generator of W(S) for a graded smooth surface should be realizable by a formal dissection, mirroring the orbifold case where formal dissections classify derived-equivalent algebras.

References

Conjecture 8.12. The class of graded gentle algebras is closed under derived equiv- alence. In particular, if S = (S, E,n) is a graded smooth surface with stops, then any formal generator of W(S) is given by a formal dissection of S.

— Partially wrapped Fukaya categories of orbifold surfaces  (2407.16358 - Barmeier et al., 2024) in Conjecture 8.12, Section 8.4

This contrasts the case of smooth surfaces, where the simplicity of the notion of formal dissection reflects the closure under derived equivalence of gentle algebras which is a result of Schröer and Zimmermann in the ungraded case and a folklore conjecture in the graded case (cf. Conjecture \ref{conjecture:skewgentle} below).

— Fukaya categories of orbifold surfaces in representation theory  (2602.17370 - Barmeier et al., 19 Feb 2026) in Section “Formal generators and graded skew-gentle algebras”