Cluster categories for marked surfaces: punctured case

Abstract

We study the cluster categories arising from marked surfaces (with punctures and non-empty boundaries). By constructing skewed-gentle algebras, we show that there is a bijection between tagged curves and string objects. Applications include interpreting dimensions of ${\mathrm{Ext}}^1$ as intersection numbers of tagged curves and Auslander-Reiten translation as tagged rotation. An important consequence is that the cluster(-tilting) exchange graphs of such cluster categories are connected.

Final version, to appear in Compositio Mathematica

References in corpus (4)

• Deformed Calabi-Yau Completions
• Decorated marked surfaces: spherical twists versus braid twists
• Decorated marked surfaces II: Intersection numbers and dimensions of Homs
• A geometric model for cluster categories of type $D_n$

Cited by in corpus (26)

• Cluster algebras and continued fractions
• A geometric model for the derived category of gentle algebras
• Decorated marked surfaces: spherical twists versus braid twists
• Density of $g$-vector cones from triangulated surfaces
• On support $\tau$-tilting graphs of gentle algebras
• Categorified canonical bases and framed BPS states
• Combinatorial cluster expansion formulas from triangulated surfaces
• $q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories
• Mapping cones in the bounded derived category of a gentle algebra
• Decorated marked surfaces II: Intersection numbers and dimensions of Homs
• Quiver Grassmannians of type extended Dynkin type D - Part 1: Schubert systems and decompositions into affine spaces
• Ginzburg algebras of triangulated surfaces and perverse schobers
• $F$-matrices of cluster algebras from triangulated surfaces
• Topological structure of spaces of stability conditions and topological Fukaya type categories
• A geometric model for the module category of a skew-gentle algebra
• On Structure of cluster algebras of geometric type, II: Green's equivalences and paunched surfaces
• Decorated Marked Surfaces: Calabi-Yau categories and related topics
• An expansion formula for type A and Kronecker quantum cluster algebras
• The braid group for a quiver with superpotential
• Laminations of punctured surfaces as $\tau$-regular irreducible components
• Computation of extension spaces for the path algebra of type $\tilde{A}(n-1,1)$ using planar curves
• Compatibility degree of cluster complexes
• Denominator vectors and dimension vectors from triangulated surfaces
• A geometric realization of silting theory for gentle algebras
• On homomorphisms and generically $\tau$-regular components for skewed-gentle algebras
• Extensions in Jacobian algebras via punctured skein relations