Classifying subcategories of a Grothendieck category via its spectral category

Let $\mathcal{A}$ be a Grothendieck category. In this paper we classify subcategories of $\mathcal{A}$ via a support notion defined by Krause \cite{Kr2024} based on the spectral category of $\mathcal{A}$. We show that this aligns with Nemman’s support \cite{Ne1992}, and thereby extends earlier classifications of subcategories of the category of modules over a commutative noetherian ring and subcategories of its derived category. Over a locally coherent category $\mathcal{A}$, we show that this support agrees with open subsets of the Ziegler topology on $\mathcal{A}$.