Torsion classes of extended Dynkin quivers over commutative rings
Osamu Iyama, Yuta Kimura
TL;DR
The paper develops a framework to classify torsion classes of path algebras $RQ$ over commutative Noetherian rings $R$, focusing on extended Dynkin quivers $Q$. It constructs and analyzes a base-change map system $\{\mathrm{r}_{\mathfrak p\mathfrak q}\}$ and the compatibility condition, establishes when $RQ$ is compatible under ($R_1$) and a finite height-2 prime hypothesis, and provides a concrete description of $\mathsf{tors}(RQ)$ as compatible families of torsion classes from the fibers $\kappa(\mathfrak p)\otimes_R RQ$, with a detailed treatment for the Kronecker quiver over Dedekind domains. The results connect torsion-class theory with cluster-tilting and real Schur-root data, and extend known Dynkin-case correspondences to extended Dynkin types via reverse filtrations and derived AR-translations. The findings yield explicit combinatorial descriptions and enable a poset-isomorphism $\mathsf{tors}(RQ) \simeq \{(\mathcal X^{\mathfrak p})_{\mathfrak p}\mid\text{compatibility} ight\}$, enhancing understanding of how torsion theories behave under base change and across fibers.
Abstract
For a Noetherian $R$-algebra $Λ$, there is a canonical inclusion $\mathsf{tors}Λ\to\prod_{\mathfrak{p}\in \mathrm{Spec} R}\mathsf{tors}(κ(\mathfrak{p})Λ)$, and each element in the image satisfies a certain compatibility condition. We call $Λ$ compatible if the image coincides with the set of all compatible elements. For example, for a Dynkin quiver $Q$ and a commutative Noetherian ring $R$ containing a field, the path algebra $RQ$ is compatible. In this paper, we prove that $RQ$ is compatible when $Q$ is an extended Dynkin quiver and $R$ is either a Dedekind domain or a Noetherian semilocal normal ring of dimension two.