A complete derived invariant and silting theory for graded gentle algebras

TL;DR

The paper settles the problem of deriving a complete invariant for graded gentle algebras by linking derived equivalences to geometric data on surface models, confirming a conjecture of Lekili and Polishchuk. It develops a comprehensive silting theory for homologically smooth graded gentle algebras, classifies when per(A) admits silting objects, and provides a precise criterion for when pre-silting objects are necessarily partial silting. A central advance is the establishment of a complete derived invariant encoded by surface invariants (W_η, σ(η), ~A(η), A(η)) and their compatibility under orientation-preserving homeomorphisms of surface models, yielding a complete invariant for triangle equivalences of partially wrapped Fukaya categories. The work also connects silting reductions to surface cuts, constructs standard-form algebras to read invariants directly from quivers, and produces infinite families of counterexamples showing that not all pre-silting objects are partial silting in finite-dimensional settings, thereby deepening the bridge between representation theory and geometric topology.

Abstract

We confirm a conjecture by Lekili and Polishchuk that the geometric invariants which they construct for homologically smooth graded (not necessarily proper) gentle algebras form a complete derived invariant. Hence, we obtain a complete invariant of triangle equivalences for partially wrapped Fukaya categories of graded surfaces with stops. A key ingredient of the proof is the full description of homologically smooth graded gentle algebras whose perfect derived categories admit silting objects. We also apply this to classify which graded gentle algebras admit pre-silting objects that are not partial silting. In particular, this allows us to construct a family of counterexamples to the question whether any pre-silting object in the derived category of a finite-dimensional algebra is partial silting.

Figures (10)

• Figure 1: The polygon cut out by (green) arcs, where the non-coloured edges with orientation are boundary segments in $\partial S$. It contains one stop on one of the boundary segments.
• Figure 2: The general local picture for relations $\alpha \beta$, where the two boundary components are not necessarily pairwise distinct.
• Figure 3: A local picture of a simple closed curve $\gamma$ crossing through polygons of an admissible dissection $\Delta$.
• Figure 4: The standard surface model of an algebra $A$ of the form $(g; m_0, \dotsc, m_u; v)$, where the admissible dissection $\Delta$ is given by the arcs in green and where the line field $\eta$ is determined by the grading of $A$. The boundary component in the middle contains $m_0$ stops.
• Figure 5: The polygon cut out by the standard admissible dissection $\Delta$ for $g=2, b=1=m_0$. The corresponding gentle algebra $A(\Delta)$ is described in \ref{['equ:An']}. In the left polygon, there are four closed curves $s_i, t_i$, for $i=1,2$, whereas the right polygon contains a single closed curve $u$, subdivided into $8$ segments $u_i$ by the green arcs in sequential order. Note that $u$ is homotopic to the (unique) boundary component.

Theorems & Definitions (97)

• Theorem 1.1: The contrapositive of Theorem \ref{['Thm:existence_of_silting']}
• Corollary 1.2
• Corollary 1.3: Corollary \ref{['Cor:existence_of_silting']}
• Theorem 1.4: Theorem \ref{['Thm:derivedeq']}
• Corollary 1.5
• Corollary 1.6: Corollary \ref{['Cor:complete']}
• Theorem 1.7: Theorem \ref{['Thm:partialsiltingmainresult']}
• Corollary 1.8: Corollary \ref{['cor:finitedimensionalgentle']}
• Definition 2.1
• Definition 2.2