Gabriel-Krull dimension and minimal atoms in Grothendieck categories

Abstract: Let $\mathcal{A}$ be a Grothendieck category. In this paper, we classify localizing subcategories of a semi-noetherian category $\mathcal{A}$ through open subsets of $\mathcal{A}\mathrm{Spec}\,\mathcal{A}$. For a semi-noetherian locally coherent category $\mathcal{A}$, we introduce a new topology on $\mathcal{A}\mathrm{Spec}\,\mathcal{A}$ and we show that it is homeomorphic to the Ziegler spectrum $\mathrm{Z_g}\mathcal{A}$. Moreover, for a locally coherent category $\mathcal{A}$, we provide a new characterization of localizing subcategories of finite type of $\mathcal{A}$. We define a dimension for objects using the preorder $\leq$ on $\mathcal{A}\mathrm{Spec}\,\mathcal{A}$, which serves as a lower bound of the Gabriel-Krull dimension. Finally, we investigate the minimal atoms of a noetherian object $M$ in $\mathcal{A}$ and establish sufficient conditions for the finiteness of the number of minimal atoms of $M$.