Indecomposable objects determined by their index in Higher Homological Algebra

Abstract

Let $\mathcal{C}$ be a 2-Calabi-Yau triangulated category, and let $\mathcal{T}$ be a cluster tilting subcategory of $\mathcal{C}$. An important result from Dehy and Keller tells us that a rigid object $c\in \mathcal{C}$ is uniquely defined by its index with respect to $\mathcal{T}$. The notion of triangulated categories extends to the notion of $(d+2)$-angulated categories. Thanks to a paper by Oppermann and Thomas, we now have a definition for cluster tilting subcategories in higher dimensions. This paper proves that under a technical assumption, an indecomposable object in a $(d+2)$-angulated category is uniquely defined by its index with respect to a higher dimensional cluster tilting subcategory. We also demonstrate an application of this result in higher dimensional cluster categories.

Revising the article based on advice from a referee received upon submitting to a journal. This serves to make the results slightly more general