[Paper Review] Classifying subcategories and the spectrum of a locally noetherian category

Abhishek Banerjee|arXiv (Cornell University)|Oct 23, 2017

Algebraic structures and combinatorial models3 citations

TL;DR

This paper introduces a classification framework for subcategories in a locally noetherian Grothendieck category $\mathcal{A}$ using subsets of its spectrum $\mathfrak{Spec}(\mathcal{A})$. By establishing a correspondence between subcategories and spectral subsets, the authors develop new tools in local algebra within $\mathcal{A}$, yielding a spectral characterization of Serre subcategories and enhancing structural understanding 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 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.

Motivation & Objective

• To establish a correspondence between subcategories of a locally noetherian Grothendieck category and subsets of its spectrum.
• To develop local algebra techniques within the category $\mathcal{A}$ that are of independent interest.
• To characterize Serre subcategories using spectral subsets, thereby refining the structure theory of $\mathcal{A}$.

Proposed method

• Utilizes the spectrum $\mathfrak{Spec}(\mathcal{A})$ as a geometric tool to classify subcategories via their associated subsets.
• Applies categorical duality and support theory to link subcategories to closed subsets of $\mathfrak{Spec}(\mathcal{A})$.
• Introduces a notion of local cohomology and associated prime support within $\mathcal{A}$ to analyze subcategories.
• Employs the Gabriel topology and localization theory to relate subcategories to ideals in the spectrum.
• Establishes a bijection between Serre subcategories and certain spectral subsets, generalizing classical results to the non-commutative setting.
• Leverages the locally noetherian property to ensure existence of enough injectives and well-behaved filtrations.

Experimental results

Research questions

• RQ1 How can subcategories of a locally noetherian Grothendieck category be classified using the spectrum $\mathfrak{Spec}(\mathcal{A})$?
• RQ2 What spectral conditions characterize Serre subcategories in $\mathcal{A}$?
• RQ3 What new local algebra tools emerge from studying $\mathcal{A}$ as a category with a spectrum?
• RQ4 How does the spectrum $\mathfrak{Spec}(\mathcal{A})$ reflect the structure of subcategories in $\mathcal{A}$?
• RQ5 Can the spectral correspondence be used to reconstruct subcategories from their support data?

Key findings

• A bijection is established between Serre subcategories of $\mathcal{A}$ and certain closed subsets of $\mathfrak{Spec}(\mathcal{A})$, providing a spectral classification.
• The spectrum $\mathfrak{Spec}(\mathcal{A})$ enables a geometric interpretation of subcategories via support and annihilator conditions.
• New local algebra techniques are developed within $\mathcal{A}$, including a notion of local cohomology and prime support, which are of independent interest.
• The paper shows that the assignment of a subcategory to its support subset yields a complete invariant under suitable conditions.
• The framework generalizes classical results from commutative algebra to the non-commutative setting of locally noetherian categories.
• The results demonstrate that the spectrum $\mathfrak{Spec}(\mathcal{A})$ controls the lattice of subcategories in $\mathcal{A}$.