Prime spectra of abelian 2-categories and categorifications of Richardson varieties

TL;DR

This paper develops a framework for prime ideals in abelian 2-categories, extending noncommutative spectra concepts, and applies it to categorify quantized coordinate rings of Richardson varieties in Kac-Moody groups.

Contribution

It introduces a noncommutative prime spectrum theory for abelian 2-categories and applies it to categorify Richardson varieties via Serre prime ideals.

Findings

Established a noncommutative prime spectrum framework for abelian 2-categories.

Linked Serre prime spectra to $ ext{Z}_+$-rings.

Constructed categorifications of Richardson varieties using Serre prime ideals.

Abstract

We describe a general framework for prime, completely prime, semiprime, and primitive ideals of an abelian 2-category. This provides a noncommutative version of Balmer's prime spectrum of a tensor triangulated category. These notions are based on containment conditions in terms of thick subcategories of an abelian category and thick ideals of an abelian 2-category. We prove categorical analogs of the main properties of noncommutative prime spectra. Similar notions, starting with Serre subcategories of an abelian category and Serre ideals of an abelian 2-category, are developed. They are linked to Serre prime spectra of $\mathbb{Z}_+$-rings. As an application, we construct a categorification of the quantized coordinate rings of open Richardson varieties for symmetric Kac-Moody groups, by constructing Serre completely prime ideals of monoidal categories of modules of the KLR algebras, and by taking Serre quotients with respect to them.