Classifying subcategories of modules over Noetherian algebras

TL;DR

This paper unifies classification theories of torsion classes over Noetherian algebras and rings, providing new bijections, embeddings, and conditions for classifying subcategories of modules in algebraic structures.

Contribution

It introduces a unified framework for classifying torsion and torsionfree classes over Noetherian algebras, extending existing theories and establishing new conditions for compatibility.

Findings

Constructs bijections and embeddings for torsion classes

Defines compatible elements and provides conditions for their characterization

Applies results to Dynkin quivers and Cambrian lattices

Abstract

The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring $R$ and a module-finite $R$-algebra $\Lambda$, we study the set $\mathsf{tors} \Lambda$ (respectively, $\mathsf{torf}\Lambda$) of torsion (respectively, torsionfree) classes of the category of finitely generated $\Lambda$-modules. We construct a bijection from $\mathsf{torf}\Lambda$ to $\prod_{\mathfrak{p}} \mathsf{torf}(\kappa(\mathfrak{p}) \otimes_R \Lambda )$, and an embedding $\Phi_{\rm t}$ from $\mathsf{tors} \Lambda$ to $\mathbb{T}_R(\Lambda):=\prod_{\mathfrak{p}} \mathsf{tors}(\kappa(\mathfrak{p}) \otimes_R \Lambda)$, where $\mathfrak{p}$ runs all prime ideals of $R$. When $\Lambda=R$, these give classifications of torsionfree classes, torsion classes and Serre subcategories of $\mathsf{mod} R$ due to Takahashi, Stanley-Wang and Gabriel. To give a description of $\mathrm{Im} \Phi_{\rm t}$, we introduce the notion of compatible elements in $\mathbb{T}_R(\Lambda)$, and prove that all elements in $\mathrm{Im} \Phi_{\rm t}$ are compatible. We give a sufficient condition on $(R, \Lambda)$ such that all compatible elements belong to $\mathrm{Im} \Phi_{\rm t}$ (we call $(R, \Lambda)$ compatible in this case). For example, if $R$ is semi-local and $\dim R \leq 1$, then $(R, \Lambda)$ is compatible. We also give a sufficient condition in terms of silting $\Lambda$-modules. As an application, for a Dynkin quiver $Q$, $(R, RQ)$ is compatible and we have a poset isomorphism $\mathsf{tors} RQ \simeq \mathrm{Hom}_{\rm poset}(\mathrm{Spec} R, \mathfrak{C}_Q)$ for the Cambrian lattice $\mathfrak{C}_Q$ of $Q$.