Telescope conjecture for t-structures over noetherian path algebras

TL;DR

This paper investigates the structure of t-structures in derived categories of path algebras over noetherian rings, providing classifications and descriptions in specific algebraic contexts.

Contribution

It proves that homotopically smashing t-structures are compactly generated and characterizes all such structures via poset homomorphisms, extending understanding of t-structures over noetherian path algebras.

Findings

Homotopically smashing t-structures are compactly generated.

Complete classification of compactly generated t-structures via poset homomorphisms.

Description of wide subcategories when the ring is regular.

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 $\mathrm{mod}(RQ)$.