Generalized negligible morphisms and their tensor ideals

TL;DR

This paper generalizes negligible morphisms in monoidal categories, explores associated tensor and thick ideals, and provides explicit trace functions and geometric descriptions in quantum and modular settings.

Contribution

It introduces a new notion of generalized negligible morphisms, studies their tensor and thick ideals, and offers explicit trace functions and geometric descriptions in specific categories.

Findings

Explicit trace functions for thick ideals when the maximal ideal is generated by one element.

Geometric description of thick ideals in quantum type A.

Proposed geometric framework for modular categories.

Abstract

We introduce a generalization of the notion of a negligible morphism and study the associated tensor ideals and thick ideals. These ideals are defined by considering deformations of a given monoidal category $\mathcal{C}$ over a local ring $R$. If the maximal ideal of $R$ is generated by a single element, we show that any thick ideal of $\mathcal{C}$ admits an explicitely given modified trace function. As examples we consider various Deligne categories and the categories of tilting modules for a quantum group at a root of unity and for a semisimple, simply connected algebraic group in prime characteristic. We prove an elementary geometric description of the thick ideals in quantum type A and propose a similar one in the modular case.