Telescope conjecture for quiver representations over artinian rings
Michal Hrbek, Enrico Sabatini
TL;DR
The paper proves a local-to-global approach for the telescope conjecture in t-structures on derived categories of quiver representations over artinian rings. It develops an injective correspondence between homotopically smashing coaisles on the global category and collections of definable coaisles on stalk categories at prime ideals, and establishes a bijection at stalks for finite quivers. Using a lifting technique and Koszul complexes, the authors show that stalk subcategories satisfy the telescope conjecture, and then glue these local results to obtain the global telescope conjecture for D(RQ) when R is artinian. The framework further extends to non-noetherian contexts, notably commutative perfect rings, yielding analogous results and broadening the class of algebras for which the telescope conjecture holds.
Abstract
Let $\mathcal{D}(RC)$ be the derived category of representations of a small category $C$ over a commutative noetherian ring $R$. We study the homotopically smashing t-structures on this category. Specifying our discussion to the stalk categories $Γ_{\mathfrak{p}}\mathcal{D}(RQ)$ for a finite quiver $Q$ and a prime ideal $\mathfrak{p}$ of $R$, we prove the telescope conjecture for the derived category of representations of finite quivers over artinian rings. More generally, we prove the same result also outside of the noetherian context, for representations of finite quivers over commutative perfect rings.