Classifying subcategories, local algebra and the Rosenberg spectrum of a locally noetherian category

Let $\mathcal{A}$ be a locally noetherian Grothendieck category. In this paper, we study subcategories of $\mathcal{A}$ using subsets of the Rosenberg spectrum $\mathfrak{S}pec(\mathcal{A})$. Along the way, we also develop results in local algebra with respect to the category $\mathcal{A}$ that we believe to be of independent interest.