The Balmer spectrum and tensor telescope conjecture for noetherian path algebras

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.