Torsion classes of extended Dynkin quivers over commutative rings

Abstract

For a Noetherian $R$-algebra $\Lambda$, there is a canonical inclusion $\mathsf{tors}\Lambda \to {\prod }_{\mathfrak{p}\in \mathrm{Spec}R}\mathsf{tors}(\kappa (\mathfrak{p})\Lambda )$, and each element in the image satisfies a certain compatibility condition. We call $\Lambda$ 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.

20 pages, we have improved many results, in particular, (i) using results from our recent paper arXiv:2504.02371, we succeeded in removing the assumption that R contains a field in main Theorem 1.2, (ii) we provided an explicit description of the map r_{pq} for the Kronecker quiver over a Dedekind domain in Section 6