Localizations for quiver Hecke algebras III

Let $R$ be a quiver Hecke algebra, and let $\mathcal{C}_{w,v}$ be the category of finite-dimensional graded $R$-module categorifying a $q$-deformation of the doubly-invariant algebra $^{N'(w)} \mathbb{C}[N] ^{N(v)}$. In this paper, we prove that the localization $\tilde{\mathcal{C}}_{w,v}$ of the category $\mathcal{C}_{w,v}$ can be obtained as the localization by right braiders arising from determinantial modules. As its application, we show several interesting properties of the localized category $\tilde{\mathcal{C}}_{w,v}$ including the right rigidity.