On the graded dual numbers, arcs, and non-crossing partitions of the integers

TL;DR

This paper develops a combinatorial model using arcs on the integer line to describe the structure of the bounded derived category of graded dual numbers, revealing lattice structures and symmetries related to non-crossing partitions.

Contribution

It introduces a novel combinatorial framework for understanding the derived category of graded dual numbers via arcs and non-crossing partitions, connecting algebraic and combinatorial structures.

Findings

Lattice of thick subcategories described by non-crossing partitions

Model relates derived categories to combinatorial arc diagrams

Comments on symmetries and exceptional collections

Abstract

We give a combinatorial model for the bounded derived category of graded modules over the dual numbers in terms of arcs on the integer line with a point at infinity. Using this model we describe the lattice of thick subcategories of the bounded derived category, and of the perfect complexes, in terms of non-crossing partitions. We also make some comments on the symmetries of these lattices, exceptional collections, and the analogous problem for the unbounded derived category.