The Balmer spectrum and tensor telescope conjecture for noetherian path algebras

TL;DR

This paper investigates the tensor triangulated category associated with a noetherian path algebra, computing its spectrum, classifying thick tensor-ideals, and proving the tensor telescope conjecture.

Contribution

It provides a detailed description of the Balmer spectrum and classifies tensor-ideals for the category, establishing the tensor telescope conjecture in this context.

Findings

Computed the Balmer spectrum of the category.

Classified all thick tensor-ideals.

Proved the tensor telescope conjecture.

Abstract

Given a commutative noetherian ring $R$ and a finite acyclic quiver $Q$, we study the tensor triangulated category $\mathcal{D}(RQ)$ endowed with the vertexwise tensor product. We find a description of the internal hom functor and show that the category is not rigid. We compute its Balmer spectrum and, despite the non-rigidity, we get a classification of all the thick tensor-ideals and a stratification result. After introducing the notion of tensor-t-structure, we give a classification of the compactly generated ones and prove the tensor telescope conjecture.