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A geometric model for the module category of a string algebra

Karin Baur, Raquel Coelho Simoes

Abstract

In this paper, we give a geometric construction of string algebras and of their module categories. Our approach uses dissections of punctured Riemann surfaces with extra data at marked points, called labels. As an application, we give a classification of support tau-tilting modules in terms of arcs in such a tiled surface. In the case when the string algebra is gentle, we recover the classification given arXiv:2004.11136.

A geometric model for the module category of a string algebra

Abstract

In this paper, we give a geometric construction of string algebras and of their module categories. Our approach uses dissections of punctured Riemann surfaces with extra data at marked points, called labels. As an application, we give a classification of support tau-tilting modules in terms of arcs in such a tiled surface. In the case when the string algebra is gentle, we recover the classification given arXiv:2004.11136.
Paper Structure (15 sections, 20 theorems, 7 equations, 40 figures)

This paper contains 15 sections, 20 theorems, 7 equations, 40 figures.

Table of Contents

  1. Background on string algebras
  2. Strings and bands
  3. Description of morphisms between string modules
  4. Irreducible morphisms and the Auslander-Reiten translate
  5. Labelled tiling algebras
  6. Tiled surfaces and labels
  7. String algebras as labelled tiling algebras
  8. The module category of a labelled tiling algebra
  9. Permissible arcs and closed curves
  10. Morphisms via crossings and intersections
  11. Pivot elementary moves
  12. Geometric interpretation of the Auslander-Reiten translate
  13. Classification of support tau-tilting modules
  14. Crossings of $\mathsf{M}$-arcs.
  15. Support $\tau$-tilting via crossings

Key Result

Theorem A

Let $A$ be a finite dimensional monomial algebra. Then $A$ is a string algebra if and only if $A$ is a labelled tiling algebra of a marked punctured surface.

Figures (40)

  • Figure 1: The types of tiles in a dissection. Figure $(i.b)$ is the special case of a tile with boundary segments and of size $3$.
  • Figure 2: The shaded triangle is an angle at $p$. A puncture $p \in \mathsf{M}$ can have a unique angle which is a self-folded triangle.
  • Figure 3: Labeled tiling of a surface.
  • Figure 4: The quiver of the tiling from Figure \ref{['fig:label-angle-red']}. The type (1) relations are indicated by dashed (red) lines.
  • Figure 5: Non-homeomorphic surfaces associated to the same string algebra.
  • ...and 35 more figures

Theorems & Definitions (74)

  • Theorem A: Theorem \ref{['thm:model-algebra']}
  • Theorem B: Theorems \ref{['thm:permissiblearcsstrings']} and \ref{['thm:pivot-move-irreducible']}
  • Theorem C: Theorem \ref{['thm:tau-tilt']}
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • ...and 64 more