Schur roots and tilting modules of acyclic quivers over commutative rings
Osamu Iyama, Yuta Kimura
TL;DR
This work generalizes the Ingalls–Thomas cluster–tilting correspondence to path algebras $RQ$ over a ring $R$ by proving isomorphisms $ extsf{Clus}(Q) o extsf{2 ext{-}silt}RQ o extsf{s ext{-}tilt}RQ$ via $M_-^R$ and $H^0$, for a finite acyclic quiver $Q$ and ring indecomposable Noetherian $R$. It extends the field-case results to a ring-theoretic setting, leveraging real Schur roots and Crawley-Boevey-type Ext formulas to connect cluster combinatorics with silting theory. The paper further shows that when $Q$ is Dynkin, the torsion classes of $ ext{mod }RQ$ are described by order-preserving maps from $ ext{Spec }R$ to clusters, enabling a global-to-local classification of torsion phenomena. Overall, the results unify cluster algebras, silting theory, and torsion theory over rings, providing a robust framework for understanding how cluster structures behave in families parameterized by $ ext{Spec }R$.
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.