交换2-范畴的素谱与Richardson簇的扩张
Prime spectra of abelian 2-categories and categorifications of Richardson varieties
摘要：
我们描述了一个一般的框架，素，完全素，半素，和本原理想的交换2-范畴。这提供了一个非交换版本的巴尔默素谱的张量三角范畴。这些概念是基于包含条件的厚子范畴的阿贝尔范畴和厚理想的阿贝尔2-范畴。我们证明了非交换素谱的主要性质的范畴类似物。类似的概念，开始与塞尔子范畴的阿贝尔范畴和塞尔理想的阿贝尔2范畴，开发。它们与$\mathbb{Z}_+$ -环的Serre素谱有关.作为应用，通过构造KLR代数的模的幺半范畴的Serre完全素理想，并对它们取Serre等价，构造了对称Kac-Moody群的开Richardson簇的量子化坐标环的范畴.
英文摘要：
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.
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