Classifying torsion pairs of Noetherian algebras

Abstract

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 $\prod_{\mathfrak{p}} \mathsf{torf}(\Lambda\otimes_R \kappa(\mathfrak{p}))$, and an embedding $\Phi$ from $\mathsf{tors} \Lambda$ to $\mathbb{T}_R(\Lambda):=\prod_{\mathfrak{p}} \mathsf{tors}(\Lambda\otimes_R \kappa(\mathfrak{p}))$, 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 $\mathrm{Im} \Phi$, we introduce the notion of compatible elements in $\mathbb{T}_R(\Lambda)$, and prove that all elements in $\mathrm{Im} \Phi$ are compatible. We give a sufficient condition on $(R, \Lambda)$ such that all compatible elements belong to $\mathrm{Im} \Phi$ (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 $\mathsf{tors} RQ \simeq \mathsf{Hom}_{\rm poset}(\mathsf{Spec} R, \mathfrak{C}_Q)$ for the Cambrian lattice $\mathfrak{C}_Q$ of $Q$.Comment: 33 pages, one subsection was moved to Appendix, some propositions were added in Section