Telescope conjecture for t-structures over noetherian path algebras

TL;DR

Let $RQ$ be the path algebra of a Dynkin quiver $Q$ over a commutative noetherian ring $R$. The paper proves that every homotopically smashing t-structure in $\\mathcal{D}(RQ)$ is compactly generated, extending the telescope conjecture to non-stable t-structures in this setting. It provides a complete classification of compactly generated t-structures via order-preserving maps $\\sigma: Spec(R)\to Aisle(\\mathcal{D}(\\mathbb{K}Q))$, realized by projecting to and gluing over stalks, thereby unifying base-change phenomena with Dynkin quiver combinatorics. When $R$ is regular, wide subcategories of $\\mathrm{mod}(RQ)$ are described by filtrations of noncrossing partitions, using lattice lifts from indecomposables over a field and yielding a field-independent, Spec$(R)$-indexed description. Overall, the work extends Neeman–AJS–Hrbek-type classifications to $RQ$-modules for Dynkin quivers and provides a coherent framework connecting local data at primes with global t-structure structure via support theory and lattice lifts.

Abstract

Let $RQ$ be the path algebra of a Dynkin quiver $Q$ over a commutative noetherian ring $R$. We show that any homotopically smashing t-structure in the derived category of $RQ$ is compactly generated. We also give a complete description of the compactly generated t-structures in terms of poset homomorphisms from the prime spectrum of the ring $\mathrm{Spec}(R)$ to the poset of filtrations of noncrossing partitions of the quiver $\mathrm{Filt}(\mathbf{Nc}(Q))$. In the case that $R$ is regular, we also get a complete description of the wide subcategories of the category $\mathrm{mod}(RQ)$.

Figures (1)

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Theorems & Definitions (78)

• Theorem I: \ref{['Main']}
• Theorem II: \ref{['Wide']}
• Lemma 1.1: Pop
• Remark 1.2
• Lemma 1.3
• proof
• Remark 1.4: Kra
• Proposition 1.5
• proof
• Remark 1.6