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

TL;DR

This paper explores the classification of subcategories in a locally noetherian Grothendieck category using the Rosenberg spectrum, and develops local algebra results relevant to this setting.

Contribution

It introduces a novel approach to classify subcategories via the Rosenberg spectrum and presents new local algebra results within this categorical framework.

Findings

Subcategories can be characterized by subsets of the Rosenberg spectrum.

New results in local algebra applicable to Grothendieck categories.

Enhanced understanding of the structure of locally noetherian categories.

Abstract

Let $\mathcal A$ be a locally noetherian Grothendieck category. In this paper, we study subcategories of $\mathcal A$ using subsets of the Rosenberg spectrum $\mathfrak Spec(\mathcal A)$. 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.