Classifying torsion pairs of Noetherian algebras.

For a commutative Noetherian ring $R$ and a module-finite $R$-algebra $\Lambda$, we study the set $\mathsf{tors} \Lambda$ of torsion classes of the category of finitely generated $\Lambda$-modules. We construct 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 $\Lambda=R$, this gives Gabriel's classification of torsion classes and Serre subcategories of $\mathsf{mod} R$. We introduce a map ${\rm r}_{\mathfrak{p}, \mathfrak{q}}$ from $\mathsf{tors}(\Lambda\otimes_R \kappa(\mathfrak{p}))$ to $\mathsf{tors}(\Lambda\otimes_R \kappa(\mathfrak{q}))$ for a pair $\mathfrak{p} \supseteq \mathfrak{q}$ of prime ideals of $R$, and call $\{ \mathcal{X}^{\mathfrak{p}} \}\in\mathbb{T}_R(\Lambda)$ compatible if ${\rm r}_{\mathfrak{p}, \mathfrak{q}}(\mathcal{X}^{\mathfrak{p}})\supseteq \mathcal{X}^{\mathfrak{q}}$ holds for such pair. We show that all elements in the image of $\Phi$ are compatible, and call $(R, \Lambda)$ compatible if the converse holds. We study when $(R, \Lambda)$ is compatible. We show that, 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 $Q$.