Classifying subcategories of modules over Noetherian algebras

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 $Q$.