A geometric model of tube categories
arXiv:1011.0743 · doi:10.1016/j.jalgebra.2012.04.009
Abstract
We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between corresponding arcs, giving a geometric interpretation of the description of an extension group in the cluster category of a tube as a symmetrized version of the extension group in the tube. We show that a similar result holds for finite dimensional representations of the linearly oriented quiver of type A-double-infinity.
15 pages, 7 figures. Discussion of maximal rigid objects and triangulations at end of Section 3. Minor corrections
References in corpus (6)
- On triangulated orbit categories
- Acyclic Calabi-Yau categories
- On the Cluster Category of a Marked Surface
- Categorification of a frieze pattern determinant
- On a triangulated category which behaves like a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon
- Cluster algebras, quiver representations and triangulated categories
Cited by in corpus (8)
- Torsion pairs in cluster tubes
- Torsion pairs and rigid objects in tubes
- Rigid Indecomposable Modules in Grassmannian Cluster Categories
- A geometric realization of tame categories
- Classification of cosilting modules in type
- Cotorsion pairs in the cluster category of a marked surface
- A geometric interpretation of the triangulated structure of m-cluster categories
- Decomposition of torsion pairs on module categories