Tropical Duality in (d+2)-Angulated Categories

Joseph Reid

doi:10.1007/s10485-020-09625-7

Abstract

Let mathscr {C} be a 2-Calabi–Yau triangulated category with two cluster tilting subcategories mathscr {T} and mathscr {U}. A result from Jørgensen and Yakimov (Sel Math (NS) 26:71–90, 2020) and Demonet et al. (Int Math Res Not 2019:852–892, 2017) known as tropical duality says that the index with respect to mathscr {T} provides an isomorphism between the split Grothendieck groups of mathscr {U} and mathscr {T}. We also have the notion of c-vectors, which using tropical duality have been proven to have sign coherence, and to be recoverable as dimension vectors of modules in a module category. The notion of triangulated categories extends to the notion of (d+2)-angulated categories. Using a higher analogue of cluster tilting objects, this paper generalises tropical duality to higher dimensions. This implies that these basic cluster tilting objects have the same number of indecomposable summands. It also proves that under conditions of mutability, c-vectors in the (d+2)-angulated case have sign coherence, and shows formulae for their computation. Finally, it proves that under the condition of mutability, the c-vectors are recoverable as dimension vectors of modules in a module category.

Highlights

Let C be a triangulated category with certain nice properties, and let K be an algebraically closed field
The notion of a cluster tilting subcategory of C is due to [4, Definition 2.2], and we can define the index with respect to a cluster tilting subcategory [10, Section 2.1]
Instrumental to everything we do here are the notions of Oppermann–Thomas cluster tilting subcategory and index

Summary

Introduction

Let C be a triangulated category with certain nice properties, and let K be an algebraically closed field. Instrumental to everything we do here are the notions of Oppermann–Thomas cluster tilting subcategory and index These definitions require a (d + 2)-angulated category as defined by Geiss et al [3]. Definition 1.2 If we have an Oppermann–Thomas cluster tilting subcategory T = add(T ) This group is the free abelian group generated by the isomorphism classes [t] of objects t ∈ T , modulo all the relations of the form [t] = [t0] + [t1] where t ∼= t0 ⊕ t1. We define mutability in the following way: Let U be a basic Oppermann–Thomas cluster tilting object of C , and let {u1, u2, . Definition 1.4 Let C be a (d + 2)-angulated category, let U be a basic Oppermann–Thomas cluster tilting object with U = add(U ). DimK Hom T (FT (t), M) is an entry in the dimension vector of M when M ∈ mod T and Theorem C shows that certain sign coherent c-vectors can be realised as dimension vectors

Definitions

Tropical Duality

Computing c-Vectors Using Tropical Duality

Proof of Theorem C

A Counterexample