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Schur roots and tilting modules of acyclic quivers over commutative rings (2504.02371v1)

Published 3 Apr 2025 in math.RT, math.AC, and math.RA

Abstract: Let $Q$ be a finite acyclic quiver and $A_Q$ the cluster algebra of $Q$. It is well-known that for each field $k$, the additive equivalence classes of support tilting $kQ$-modules correspond bijectively with the clusters of $A_Q$. The aim of this paper is to generalize this result to any ring indecomposable commutative Noetherian ring $R$, that is, the additive equivalence classes of 2-term silting complexes of $RQ$ correspond bijectively with the clusters of $A_Q$. As an application, for a Dynkin quiver $Q$, we prove that the torsion classes of $\mathrm{mod} RQ$ corresponds bijectively with the order preserving maps from $\mathrm{Spec} R$ to the set of clusters.

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